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Primes p such that p does not divide any term of the Apéry-like sequence A000172 (also known as Type I primes).
+20
26
3, 11, 17, 19, 43, 83, 89, 97, 113, 137, 139, 163, 193, 211, 233, 241, 283, 307, 313, 331, 347, 353, 379, 401, 409, 419, 433, 443, 491, 499, 523, 547, 569, 587, 601, 617, 619, 641, 643, 673, 811, 827, 859, 881, 929, 947, 953, 977, 1009, 1019, 1033, 1049, 1051
COMMENTS
See Schulte et al. (2014) for the precise definition of Type I primes.
MATHEMATICA
maxPrime = 1051;
maxPi = PrimePi @ maxPrime;
okQ[p_] := AllTrue[Range[3 maxPi (* coeff 3 is empirical *)], GCD[HypergeometricPFQ[{-#, -#, -#}, {1, 1}, -1], p] == 1&];
CROSSREFS
For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see this sequence, A291275- A291284 and A133370 respectively.
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
+20
15
1, 2, 7, 30, 147, 786, 4472, 26644, 164477, 1044258, 6782484, 44887236, 301782361, 2056250570, 14172792355, 98667874038, 692948001906, 4904403499992, 34951124337300, 250617829087656, 1807055528439771, 13095146839953030
COMMENTS
Analogous to the square of the g.f. of Catalan numbers ( A000108): C(x)^2 = exp( Sum_{n>=1} A000984(n)*x^n/n ) where central binomial coefficient A000984(n) = Sum_{k=0..n} C(n,k)^2.
FORMULA
a(n) ~ c * 8^n / n^2, where c = 0.58462945... - Vaclav Kotesovec, Nov 27 2017, updated Oct 29 2024
EXAMPLE
G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +...
log(A(x)) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
MATHEMATICA
a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/n, {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *) nmax = 30; Clear[a]; franel = RecurrenceTable[{n^2*a[n] == (7*n^2 - 7*n + 2)*a[n-1] + 8*(n-1)^2*a[n-2], a[1] == 2, a[2] == 10}, a, {n, 1, nmax}]; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[franel[[k]]*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 27 2024 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3)*x^m/m)+x*O(x^n)), n)}
Least prime divisor of Fr(n) which does not divide any Fr(k) with k < n, or 1 if such a primitive prime divisor of Fr(n) does not exist, where Fr(n) denotes the n-th Franel number given by A000172.
+20
11
2, 5, 7, 173, 563, 13, 41, 369581, 937, 61, 23, 29, 2141, 12148537, 31, 157, 59, 37, 506251, 151, 3019, 769, 47, 6730949, 79, 53, 3853, 661, 138961158000728258971, 1361, 421, 96920594213, 51378681049, 457, 71
COMMENTS
Conjecture: a(n) > 1 for all n > 0.
EXAMPLE
a(7) = 41 since Fr(7) = 2^9*5*41 with the prime factor 41 dividing none of Fr(1), ..., Fr(6) but 2 divides Fr(1) = 2 and 5 divides Fr(2) = 10.
MATHEMATICA
Fr[n_]:=Sum[Binomial[n, k]^3, {k, 0, n}]
f[n_]:=FactorInteger[Fr[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[Do[Do[If[Mod[Fr[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 35}]
G.f. A(x) satisfies: the sum of the coefficients of x^k, k=0..n, in A(x)^n equals Sum_{k=0..n} C(n,k)^3 = A000172(n) (Franel numbers), for n>=0.
+20
9
1, 1, 3, 7, 20, 66, 244, 980, 4182, 18674, 86353, 410541, 1996214, 9888844, 49760925, 253767097, 1309154825, 6822023553, 35865392690, 190038440422, 1014015337209, 5444707218851, 29401289997403, 159584901816255, 870267544114291, 4766246752344215, 26206635040151511
COMMENTS
Compare to: Sum_{k=0..n} [x^k] 1/(1-x)^n = Sum_{k=0..n} C(n,k)^2 = (2*n)!/n!^2.
FORMULA
Given g.f. A(x), Sum_{k=0..n} [x^k] A(x)^n = Sum_{k=0..n} C(n,k)^3 = A000172(n). Given g.f. A(x), let G(x) = A(x*G(x)) then (G(x) + x*G'(x)) / (G(x) - x*G(x)^2) = Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)/(1-2*x)^(3*n+1) = Sum_{n>=0} A000172(n)*x^n.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 20*x^4 + 66*x^5 + 244*x^6 + 980*x^7 +...
ILLUSTRATION OF INITIAL TERMS.
If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
A^0: [1],0, 0, 0, 0, 0, 0, 0, 0, ...;
A^1: [1, 1], 3, 7, 20, 66, 244, 980, 4182, ...;
A^2: [1, 2, 7], 20, 63, 214, 789, 3124, 13112, ...;
A^3: [1, 3, 12, 40], 138, 492, 1848, 7326, 30531, ...;
A^4: [1, 4, 18, 68, 255], 960, 3716, 14920, 62295, ...;
A^5: [1, 5, 25, 105, 425, 1691], 6785, 27805, 117165, ...;
A^6: [1, 6, 33, 152, 660, 2772, 11560], 48588, 207774, ...;
A^7: [1, 7, 42, 210, 973, 4305, 18676, 80746],351792, ...;
A^8: [1, 8, 52, 280, 1378, 6408, 28916, 128808, 573311], ...; ...
then the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals Sum_{k=0..n} C(n,k)^3 = A000172(n): A000172(4) = 1 + 4 + 18 + 68 + 255 = 346;
A000172(5) = 1 + 5 + 25 + 105 + 425 + 1691 = 2252;
A000172(6) = 1 + 6 + 33 + 152 + 660 + 2772 + 11560 = 15184; ...
MATHEMATICA
Franel[n_] := Sum[Binomial[n, k]^3, {k, 0, n}];
a[0] = 1; a[n_] := Module[{B, G}, B = Sum[Franel[k]*x^k, {k, 0, n+1}] + x^3*O[x]^n; G = 1+x*O[x]^n; For[i=1, i <= n, i++, G = 1+Integrate[(B-1)* (G/x)-B*G^2, x]]; SeriesCoefficient[x/InverseSeries[x*G, x], {x, 0, n}]];
PROG
(PARI) /* By Definition (slow): */
{Franel(n)=sum(k=0, n, binomial(n, k)^3)}
{a(n)=if(n==0, 1, (Franel(n) - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Faster, using series reversion: */
{Franel(n)=sum(k=0, n, binomial(n, k)^3)}
{a(n)=local(B=sum(k=0, n+1, Franel(k)*x^k)+x^3*O(x^n), G=1+x*O(x^n));
for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); polcoeff(x/serreverse(x*G), n)}
for(n=0, 30, print1(a(n), ", "))
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/(2*n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
+20
6
1, 1, 3, 12, 57, 300, 1693, 10045, 61890, 392688, 2550843, 16891566, 113660475, 775223595, 5349057132, 37280705406, 262119009927, 1857241951359, 13250054817027, 95110710932424, 686490953423700, 4979704242810870
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1693*x^6 +...
log(A(x)^2) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
MATHEMATICA
a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/(2*n), {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3)/2*x^m/m)+x*O(x^n)), n)}
Least positive integer m such that m + n divides f(m) + f(n), where f(.) is given by A000172.
+20
6
1, 1, 25, 6, 14, 4, 13, 49, 19, 10, 2, 56, 2, 5, 6, 5, 27, 61, 9, 33, 23, 53, 21, 15, 3, 24, 11, 58, 39, 118, 3, 1598, 20, 40, 4, 2, 58, 26, 29, 17, 47, 34, 4, 31, 43, 163, 41, 25, 8, 26, 67, 40, 21, 214, 535, 12, 7, 22, 164, 74
COMMENTS
Conjecture: a(n) exists for any n > 0.
EXAMPLE
a(5) = 14 since 5 + 14 = 19 divides f(5) + f(14) = 2252 + 112738423360 = 112738425612 = 19*5933601348.
MATHEMATICA
f[n_]:=Sum[Binomial[n, k]^3, {k, 0, n}]
Do[m=1; Label[aa]; If[Mod[f[m]+f[n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^n*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
+20
5
1, 2, 52, 58640, 3583098592, 11584364000042912, 2042518153012624794424576, 20047892010468651075834167466942080, 11138509206681372983092694151616405935206616064, 354938139483847646086359348765071470756626699510545192807936
EXAMPLE
G.f.: A(x) = 1 + 2*x + 52*x^2 + 58640*x^3 + 3583098592*x^4 +...
where
log(A(x)) = 2*x + 10^2*x^2/2 + 56^3*x^3/3 + 346^4*x^4/4 + 2252^5*x^5/5 + 15184^6*x^6/6 + 104960^7*x^7/7 +...+ A000172(n)^n*x^n/n +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3)^m*x^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
1, 4, 100, 3136, 119716, 5071504, 230553856, 11016601600, 546360462244, 27888242788624, 1456587070867600, 77515424509446400, 4189899499315360000, 229472379264509977600, 12709952101698593689600, 710863065714510068187136
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.
FORMULA
P-recursive: P(n-1)*n^4*a(n) = P(n)*Q(n)*a(n-1) + 8*P(n-1)*Q(n)*a(n-2) - 512*P(n)*(n-2)^4*a(n-3), where P(n) = 7*n^2 - 7*n + 2 and Q(n) = 57*n^4 - 228*n^3 + 321*n^2 - 186*n + 40 with a(0) = 1, a(1) = 4 and a(2) = 100. - Peter Bala, Feb 01 2024
MATHEMATICA
A052144[n_] := HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]^2;
a(n) = A000984(n)* A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.
+20
4
1, 4, 60, 1120, 24220, 567504, 14030016, 360222720, 9513014940, 256758913840, 7051260776560, 196403499277440, 5535202897806400, 157551884911456000, 4522682234563776000, 130783762623673221120, 3806221127760278029980
COMMENTS
Diagonal of the rational function R(x,y,z,w)=1/(1-(w*x*y+w*z+x*y+x*z+y+z)). - Gheorghe Coserea, Jul 13 2016
FORMULA
a(n) = C(2n,n) * Sum_{k=0..n} C(n,k)^3.
E.g.f.: Sum_{n>=0} a(n)*x^n/(n!*(2*n)!) = ( Sum_{n>=0} x^n/n!^3 )^2.
1/Pi
= (2/25)*Sum_{n>=0} a(n)*(9n+2)/50^n. [Cooper, equation (5)]
G.f.: 4*hypergeom([1/6, 1/3],[1],(27/2)*(1+(1-32*x)^(1/2))*(1-(1-32*x)^(1/2))^2/(3+(1-32*x)^(1/2))^3)^2/(3+(1-32*x)^(1/2)). - Mark van Hoeij, May 07 2013 Recurrence: n^3*a(n) = 2*(2*n-1)*(7*n^2 - 7*n + 2)*a(n-1) + 32*(n-1)*(2*n-3)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014 0 = (-x^2+28*x^3+128*x^4)*y''' + (-3*x+126*x^2+768*x^3)*y'' + (-1+92*x+864*x^2)*y' + (4+96*x)*y, where y is g.f. - Gheorghe Coserea, Jul 13 2016
EXAMPLE
E.g.f.: A(x) = 1 + 4*x/2! + 60*x^2/(2!*4!) + 1120*x^3/(3!*6!) + 24220*x^4/(4!*8!) + 567504*x^5/(5!*10!) +....
where A(x)^(1/2) = 1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 +x^5/5!^3 +...
MATHEMATICA
Table[Binomial[2n, n]*Sum[Binomial[n, k]^3, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)
PROG
(PARI) {a(n)=binomial(2*n, n)*sum(k=0, n, binomial(n, k)^3)}
(PARI) {a(n)=(2*n)!*n!*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^2, n)}
G.f.: A(x) = exp( Sum_{n>=1} A000172(n)^2*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
+20
4
1, 4, 58, 1256, 35771, 1200188, 45016678, 1827941560, 78753548245, 3551810922324, 166120394053698, 8002733850225288, 395089619067741926, 19911864121386482264, 1021345223473335336668, 53190166903606336969840, 2807000233813092463820488, 149869216802426305919295328
EXAMPLE
G.f.: A(x) = 1 + 4*x + 58*x^2 + 1256*x^3 + 35771*x^4 + 1200188*x^5 +...
such that
log(A(x)) = 4*x + 100*x^2/2 + 3136*x^3/3 + 119716*x^4/4 + 5071504*x^5/5 +...+ A000172(n)^2*x^n/n +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3)^2*x^m*1^m/m+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))
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