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a(n) = (2*n+4)!!/4!!, related to A000165 (even double factorials).
+20
13
1, 6, 48, 480, 5760, 80640, 1290240, 23224320, 464486400, 10218700800, 245248819200, 6376469299200, 178541140377600, 5356234211328000, 171399494762496000, 5827582821924864000, 209792981589295104000, 7972133300393213952000, 318885332015728558080000
COMMENTS
Row m=4 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
FORMULA
a(n) = (2*n+4)!!/4!!.
E.g.f.: 1/(1-2*x)^3.
a(n) ~ 2^(-1/2)*Pi^(1/2)*n^(5/2)*2^n*e^-n*n^n*{1 + 37/12*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n+1) = (2*n + 6)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 6*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 6*x/(1 - 2*x/(1 - 8*x/(1 - 4*x/(1 - 10*x/(1 - 6*x/(1 - ... - (2*n + 4)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 6*x/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
Sum_{n>=0} 1/a(n) = 8*sqrt(e) - 12.
Sum_{n>=0} (-1)^n/a(n) = 8/sqrt(e) - 4. (End)
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, 2*(n+2)*a(n-1)) end:
PROG
(PARI) vector(21, n, 2^(n-2)*(n+1)! ) \\ G. C. Greubel, Nov 11 2019 (Magma) [2^(n-1)*Factorial(n+2): n in [0..20]]; // G. C. Greubel, Nov 11 2019 (Sage) [2^(n-1)*factorial(n+2) for n in (0..20)] # G. C. Greubel, Nov 11 2019 (GAP) List([0..20], n-> 2^(n-1)*Factorial(n+2) ); # G. C. Greubel, Nov 11 2019
CROSSREFS
Cf. A052587 (essentially the same).
a(n) = (2*n+6)!!/6!!, related to A000165 (even double factorials).
+20
11
1, 8, 80, 960, 13440, 215040, 3870720, 77414400, 1703116800, 40874803200, 1062744883200, 29756856729600, 892705701888000, 28566582460416000, 971263803654144000, 34965496931549184000, 1328688883398868992000
COMMENTS
Row m=6 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
FORMULA
a(n) = (2*n+6)!!/6!!.
E.g.f.: 1/(1-2*x)^4.
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+8)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013 a(n+1) = (2*n + 8)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 8*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 8*x/(1 - 2*x/(1 - 10*x/(1 - 4*x/(1 - 12*x/(1 - 6*x/(1 - ... - (2*n + 6)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 8*x/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
Sum_{n>=0} 1/a(n) = 48*sqrt(e) - 78.
Sum_{n>=0} (-1)^n/a(n) = 30 - 48/sqrt(e). (End)
MATHEMATICA
Table[2^n*Pochhammer[4, n], {n, 0, 20}] (* G. C. Greubel, Nov 11 2019 *)
PROG
(PARI) vector(20, n, prod(j=1, n-1, 2*j+6) ) \\ G. C. Greubel, Nov 11 2019 (Magma) [1] cat [(&*[2*j+6: j in [1..n]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019 (Sage) [product( (2*j+6) for j in (1..n)) for n in (0..20)] # G. C. Greubel, Nov 11 2019 (GAP) List([0..20], n-> Product([1..n], j-> 2*j+6) ); # G. C. Greubel, Nov 11 2019
a(n) = (2*n+8)!!/8!!, related to A000165 (even double factorials).
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10
1, 10, 120, 1680, 26880, 483840, 9676800, 212889600, 5109350400, 132843110400, 3719607091200, 111588212736000, 3570822807552000, 121407975456768000, 4370687116443648000, 166086110424858624000, 6643444416994344960000
COMMENTS
Row m=8 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.
FORMULA
a(n) = (2*n+8)!!/8!!.
E.g.f.: 1/(1-2*x)^5.
a(n+1) = (2*n + 10)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 10*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 10*x/(1 - 2*x/(1 - 12*x/(1 - 4*x/(1 - 14*x/(1 - 6*x/(1 - ... - (2*n + 8)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 10*x/(1 - 12*x/(1 - 2*x/(1 - 14*x/(1 - 4*x/(1 - 16*x/(1 - 6*x/(1 - ... - (2*n + 10)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
Sum_{n>=0} 1/a(n) = 384*sqrt(e) - 632.
Sum_{n>=0} (-1)^n/a(n) = 384/sqrt(e) - 232. (End)
MATHEMATICA
Table[2^n*Pochhammer[5, n], {n, 0, 20}] (* G. C. Greubel, Nov 12 2019 *)
PROG
(PARI) vector(20, n, n--; (n+4)!*2^(n-1)/12) \\ Michel Marcus, Feb 09 2015 (Magma) F:=Factorial; [2^n*F(n+4)/F(4): n in [0..20]]; // G. C. Greubel, Nov 12 2019 (Sage) f=factorial; [2^n*f(n+4)/f(4) for n in (0..20)] # G. C. Greubel, Nov 12 2019 (GAP) F:=Factorial;; List([0..20], n-> 2^n*F(n+4)/F(4) ); # G. C. Greubel, Nov 12 2019
Decimal expansion of the number whose Pierce expansion has the sequence of double factorial numbers ( A000165) as coefficients.
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5
3, 5, 2, 8, 0, 6, 4, 3, 8, 1, 0, 6, 6, 5, 0, 0, 3, 6, 4, 6, 2, 1, 2, 3, 6, 0, 5, 3, 1, 0, 7, 3, 0, 0, 8, 6, 3, 1, 1, 1, 4, 5, 9, 6, 9, 4, 4, 4, 9, 9, 0, 1, 7, 4, 0, 2, 7, 4, 9, 4, 6, 3, 1, 0, 7, 1, 8, 6, 4, 7, 0, 1, 5, 3, 3, 6, 5, 6, 5, 4, 4, 1, 4, 5, 6, 9, 0, 9, 1, 8, 9, 6, 0, 9, 4, 8, 3, 3, 9
MAPLE
P:=proc(n) local a, i, j, k, w; a:=0; w:=1; for i from 0 by 1 to n do k:=i; j:=i-2; while j>0 do k:=k*j; j:=j-2; od; if (i=0 or i=1) then k:=1; fi; if i=2 then k:=2; fi; w:=w*k; a:=a+(-1)^i/w; print(evalf(a, 100)); od; end: P(100);
MATHEMATICA
RealDigits[N[(Sum[(-1)^n*Product[1/((k + 1)!!), {k, 1, n}], {n, 1, 250}]), 100]][[1]] (* G. C. Greubel, Jan 01 2017 *)
Decimal expansion of the inverse of the number whose Engel expansion has the sequence of double factorial numbers ( A000165) as coefficients.
+20
5
3, 7, 1, 8, 9, 6, 7, 8, 6, 2, 4, 4, 2, 5, 5, 8, 4, 7, 8, 3, 9, 5, 5, 1, 5, 3, 1, 1, 0, 6, 8, 3, 4, 0, 0, 3, 3, 4, 4, 1, 4, 2, 1, 6, 5, 0, 6, 7, 9, 1, 3, 0, 0, 2, 2, 8, 1, 1, 2, 5, 3, 9, 1, 1, 3, 8, 9, 3, 4, 8, 3, 0, 4, 4, 4, 1, 7, 6, 7, 7, 6, 4, 3, 0, 9, 3, 0, 2, 6, 3, 3, 1, 0, 7, 2, 5, 3, 6, 5
MAPLE
P:=proc(n) local a, i, j, k, w; a:=0; w:=1; for i from 0 by 1 to n do k:=i; j:=i-2; while j>0 do k:=k*j; j:=j-2; od; if (i=0 or i=1) then k:=1; fi; if i=2 then k:=2; fi; w:=w*k; a:=a+1/w; print(evalf(1/a, 100)); od; end: P(100);
MATHEMATICA
RealDigits[N[(1/Sum[Product[1/((k - 1)!!), {k, 1, n}], {n, 1, 250}]), 100]][[1]] (* G. C. Greubel, Jan 01 2016 *)
1, 2, 6, 2, 24, 24, 120, 240, 24, 720, 2400, 720, 5040, 25200, 15120, 720, 40320, 282240, 282240, 40320, 362880, 3386880, 5080320, 1451520, 40320, 3628800, 43545600, 91445760, 43545600, 3628800, 39916800, 598752000, 1676505600, 1197504000, 199584000, 3628800
COMMENTS
Construct the infinite array of polynomials
a(0,t) = 1
a(1,t) = 2
a(2,t) = 6 + 2 t
a(3,t) = 24 + 24 t
a(4,t) = 120 + 240 t + 24 t^2
a(5,t) = 720 + 2400 t + 720 t^2
a(6,t) = 5040 + 25200 t + 15120 t^2 + 720 t^3
This array is the reciprocal array of the following array b(n,t) under the list partition transform and its associated operations described in A133314. b(0,t) = 1, b(1,t) = -2, b(2,t) = -2*(t-1), b(n,t) = 0 for n>2.
Lower triangular matrix A110327 = binomial(n,k)*a(n-k,2). A110330 = matrix inverse of binomial(n,k)*a(n-k,2) = binomial(n,k)*b(n-k,2).
Let {P(n,x)}n>=0 be a polynomial sequence. Koutras has defined generalized Eulerian numbers associated with the sequence P(n,x) as the coefficients A(n,k) in the expansion of P(n,x) in a series of factorials of degree n, namely P(n,x) = Sum_{k=0..n} A(n,k)* binomial(x+n-k,n). The choice P(n,x) = x^n produces the classical Eulerian numbers of A008292. Let now P(n,x) = x*(x + 1)*...*(x + n - 1) denote the n-th rising factorial polynomial. Then the present table is the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x). See A228955 for the generalized Eulerian numbers associated with the polynomial sequence P(n,2*x + 1). (End)
FORMULA
E.g.f. for the polynomials b(.,t), introduced above, is 1 - 2x - (t-1) * x^2; therefore e.g.f. for the polynomials a(.,t), which are the row polynomials of this array, is 1 / ( 1 - 2x - (t-1) * x^2 ) = (t-1) / ( t - ( 1 + x*(t-1) )^2 ).
Also, a(n,t) = (1 - t*u^2)^(n+1) (D_u)^n [ 1 / (1 - t*u^2) ] with eval. at u = 1/t. Compare A076743. a(n,t) = n!*Sum_{k>=0} binomial(n+1,2k+1) * t^k = n!*Sum_{k>=0} A034867(n,k) * t^k. Additional relations are given by formulas in A133314. Recurrence equation: T(n+1,k) = (n+2 +2*k)T(n,k) + (n +2 -2*k)T(n,k-1).
Let P(n,x) = x*(x + 1)*...*(x + n - 1) denote the n-th rising factorial.
T(n,k) = Sum_{j=0..k+1} (-1)^(k+1-j)*binomial(n+1,k+1-j)*P(n,2*j) for n >= 1.
The row polynomial a(n,t) satisfies t*a(n,t)/(1 - t)^(n+1) = Sum_{j>=1} P(n,2*j)*t^j. For example, for n = 3 we have t*(24 + 24*t)/(1 - t)^4 = 2*3*4*t + (4*5*6)*t^2 + (6*7*8)*t^3 + ..., while for n = 4 we have t*(120 + 240*t + 24*t^2)/(1 - t)^5 = (2*3*4*5)*t + (4*5*6*7)*t^2 + (6*7*8*9)*t^3 + .... (End)
EXAMPLE
Triangle begins as:
1;
2;
6, 2;
24, 24;
120, 240, 24;
720, 2400, 720;
5040, 25200, 15120, 720;
40320, 282240, 282240, 40320;
362880, 3386880, 5080320, 1451520, 40320;
3628800, 43545600, 91445760, 43545600, 3628800;
MAPLE
for n from 0 to 10 do
seq( n!*binomial(n+1, 2*k+1), k = 0..floor(n/2) )
MATHEMATICA
Table[n!*Binomial[n+1, 2*k+1], {n, 0, 10}, {k, 0, Floor[n/2]}]//Flatten (* G. C. Greubel, Dec 30 2019 *)
PROG
(PARI) T(n, k) = n!*binomial(n+1, 2*k+1);
for(n=0, 10, for(k=0, n\2, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 30 2019 (Magma) [Factorial(n)*Binomial(n+1, 2*k+1): k in [0..Floor(n/2)], n in [0..10]]; // G. C. Greubel, Dec 30 2019 (Sage) [[factorial(n)*binomial(n+1, 2*k+1) for k in (0..floor(n/2))] for n in (0..10)] # G. C. Greubel, Dec 30 2019 (GAP) Flat(List([0..10], n-> List([0..Int(n/2)], k-> Factorial(n)*Binomial(n+1, 2*k+1) ))); # G. C. Greubel, Dec 30 2019
EXTENSIONS
Removed erroneous and duplicate statements. - Tom Copeland, Dec 03 2013
Triangle by rows, generated from the odd integers and related to A000165.
+20
3
1, 1, 1, 4, 3, 1, 24, 18, 5, 1, 192, 144, 40, 7, 1, 1920, 1440, 400, 70, 9, 1, 23040, 17280, 4800, 840, 108, 11, 1, 322560, 241920, 67200, 11760, 1512, 154, 13, 1, 5160960, 3870720, 1075200, 188160, 24192, 2464, 208, 15, 1
COMMENTS
Row sums = A000165, the double factorial numbers: (1, 2, 8, 48, 384,...). Left border = A002866 and the eigensequence of the odd integers prefaced with a 1.
FORMULA
Eigentriangle of triangle A158405 (odd integers in every row: (1, 3, 5,...)); the inverse of: 1;
-1, 1;
-1, -3, 1;
-1, -3, -5, 1;
-1, -3, -5, -7, 1;
...
EXAMPLE
First few rows of the triangle:
1;
1, 1;
4, 3, 1;
24, 18, 5, 1;
192, 144, 40, 7, 1;
1920, 1440, 400, 70, 9, 1;
23040, 17280, 4800, 840, 108, 11, 1;
322560, 241920, 67200, 11760, 1512, 154, 13, 1;
...
MAPLE
T:= proc(n) option remember; local M;
M:= (Matrix(n+1, (i, j)-> `if`(i=j, 1, `if`(i>j, -2*j+1, 0)))^(-1));
seq(M[n+1, k], k=1..n+1)
end:
MATHEMATICA
T[n_] := T[n] = Module[{M}, M = Table[If[i == j, 1, If[i>j, -2*j+1, 0]], {i, 1, n+1 }, {j, 1, n+1}] // Inverse; M[[n+1]]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)
Hankel transform of double factorial numbers n!*2^n= A000165(n).
+20
2
1, 4, 256, 589824, 86973087744, 1282470362637926400, 2723154477021188283432960000, 1133321924829207204666583887642624000000, 120746421332702772771144114237731253721340313600000000
COMMENTS
By the properties of the Hankel transform, a(n)=2^(n(n+1))* A055209(n).
FORMULA
a(n) = Product_{k=1..n} (2k)^(2(n-k+1)).
a(n) ~ 2^((n+1)^2) * Pi^(n+1) * n^(n^2 + 2*n + 5/6) / (A^2 * exp(3*n^2/2 + 2*n - 1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019
MATHEMATICA
Table[Product[(2k)^(2(n-k+1)), {k, n}], {n, 0, 10}] (* Harvey P. Dale, Apr 11 2013 *)
PROG
(PARI) for(n=0, 10, print1(prod(k=1, n, (2*k)^(2*(n-k+1))), ", ")) \\ G. C. Greubel, Oct 14 2018 (Magma) [1] cat [(&*[(2*k)^(2*(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018
Numbers n such that n!*2^n + n + 1 = A000165(n) + n + 1 is prime.
+20
1
COMMENTS
For comparison, A000165(n)+1 is prime for n = 0, 1, 259, 16708, 18655,... ( A256594).
EXAMPLE
n=4, 2^4*4! + 4 + 1 = 389 is prime.
MATHEMATICA
Select[Range[100], PrimeQ[#! 2^# + # + 1] &] (* Giovanni Resta, May 31 2018 *)
Prime values of n!*2^n+n+1 = A000165(n)+n+1.
+20
1
2, 11, 389, 1961990553613, 1678343852714360832019, 25563186766285862273530264901662157745369907200000037
COMMENTS
For a(6), n = 36. There are no additional terms up to n=1000. - Harvey P. Dale, Aug 06 2012 Primes for n = 0, 2, 4, 12, 18, 36, and no others for n < 5001. - Robert G. Wilson v, Aug 07 2012
EXAMPLE
If n=4, 2^4*4!+4+1 = 389 is prime, so 389 is a term.
MATHEMATICA
Select[Table[n! 2^n+n+1, {n, 0, 1000}], PrimeQ] (* Harvey P. Dale, Aug 06 2012 *)
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