A119900 -id:A119900

Search: a119900 -id:a119900

Displaying 11-14 of 14 results found.

page 1 2

A207543

Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

+10
3

1, 0, 3, 0, 1, 5, 0, 0, 5, 7, 0, 0, 1, 14, 9, 0, 0, 0, 7, 30, 11, 0, 0, 0, 1, 27, 55, 13, 0, 0, 0, 0, 9, 77, 91, 15, 0, 0, 0, 0, 1, 44, 182, 140, 17, 0, 0, 0, 0, 0, 11, 156, 378, 204, 19, 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Previous name was: "A scaled version of triangle A082985."

Triangle, read by rows, given by (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

LINKS

Table of n, a(n) for n=0..66.

R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

FORMULA

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 3.

G.f.: (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A078069(n), A000007(n), A003945(n), A111566(n) for x = -1, 0, 1, 2 respectively.

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A090131(n), A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively.

From Peter Bala, Aug 17 2016: (Start)

Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:

Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.

For example, for row 6 we find S(6,n) + 27*S(8,n) + 55*S(10,n) + 13*S(12,n) = 1/2*(2*n + 1)*(n*(n + 1))^6.

There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)

EXAMPLE

Triangle begins :

0, 3

0, 1, 5

0, 0, 5, 7

0, 0, 1, 14, 9

0, 0, 0, 7, 30, 11

0, 0, 0, 1, 27, 55, 13

0, 0, 0, 0, 9, 77, 91, 15

0, 0, 0, 0, 1, 44, 182, 140, 17

0, 0, 0, 0, 0, 11, 156, 378, 204, 19

0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21

0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23

MAPLE

s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s, x, 12):

seq(print(seq(coeff(coeff(t, x, n), y, m), m=0..n)), n=0..11); # Peter Luschny, Aug 17 2016

CROSSREFS

Cf. A082985 which is another version of this triangle.

Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Feb 24 2012

EXTENSIONS

New name using a formula of the author from Peter Luschny, Aug 17 2016

STATUS

approved

A109447

Binomial coefficients C(n,k) with n-k odd, read by rows.

+10
2

1, 2, 1, 3, 4, 4, 1, 10, 5, 6, 20, 6, 1, 21, 35, 7, 8, 56, 56, 8, 1, 36, 126, 84, 9, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 12, 220, 792, 792, 220, 12, 1, 78, 715, 1716, 1287, 286, 13, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435, 3003, 455, 15

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The same as A119900 without 0's. A reflected version of A034867 or A202064. - Alois P. Heinz, Feb 07 2014

From Vladimir Shevelev, Feb 07 2014: (Start)

Also table of coefficients of polynomials P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x). The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)

LINKS

Alois P. Heinz, Rows n = 1..200, flattened

EXAMPLE

Starred terms in Pascal's triangle (A007318), read by rows:

1*, 1;

1, 2*, 1;

1*, 3, 3*, 1;

1, 4*, 6, 4*, 1;

1*, 5, 10*, 10, 5*, 1;

1, 6*, 15, 20*, 15, 6*, 1;

1*, 7, 21*, 35, 35*, 21, 7*, 1;

1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;

1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;

Triangle T(n,k) begins:

1, 3;

4, 4;

1, 10, 5;

6, 20, 6;

1, 21, 35, 7;

8, 56, 56, 8;

1, 36, 126, 84, 9;

10, 120, 252, 120, 10;

MAPLE

T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):

seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20); # Alois P. Heinz, Feb 07 2014

MATHEMATICA

Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

CROSSREFS

Cf. A109446.

KEYWORD

easy,nonn,tabf

AUTHOR

Philippe Deléham, Aug 27 2005

EXTENSIONS

More terms from Robert G. Wilson v, Aug 30 2005

Corrected offset by Alois P. Heinz, Feb 07 2014

STATUS

approved

A202064

Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

+10
1

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Riordan array (x/(1-x)^2, x^2/(1-x)^2).

Mirror image of triangle in A119900.

A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011

LINKS

Table of n, a(n) for n=0..65.

FORMULA

G.f.: 1/((1-x)^2-y*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.

T(n,k) = binomial(n+1,2k+1).

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

EXAMPLE

Triangle begins :

2, 0

3, 1, 0

4, 4, 0, 0

5, 10, 1, 0, 0

6, 20, 6, 0, 0, 0

7, 35, 21, 1, 0, 0, 0

8, 56, 56, 8, 0, 0, 0, 0

CROSSREFS

Cf. A007318, A000079 (row sums), A005314 (antidiagonal sums), A119900, A034867, A084938, A130595, A203322.

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Dec 10 2011

STATUS

approved

A265644

Triangle read by rows: T(n,m) is the number of quaternary words of length n with m strictly increasing runs (0 <= m <= n).

+10
1

1, 0, 4, 0, 6, 10, 0, 4, 40, 20, 0, 1, 65, 155, 35, 0, 0, 56, 456, 456, 56, 0, 0, 28, 728, 2128, 1128, 84, 0, 0, 8, 728, 5328, 7728, 2472, 120, 0, 0, 1, 486, 8451, 27876, 23607, 4950, 165

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

In the following description the alphabet {0..r} is taken as a basis, with r = 3 in this case.

For example, the quaternary word 2|03|123|3 of length n=7, has m=4 strictly increasing runs.

The empty word has n = 0 and m = 0, and T(0, 0) = 1.

T(n, 0) = 0 for n >= 1.

T(n, m) <> 0 for m <= n <= m*(r+1). T(m*(r+1), m) = 1.

T(n,m) is a partition, based on m, of all the words of length n, so Sum_{k=0..n} T(n,k) = (r+1)^n.

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 24, p. 154.

LINKS

Table of n, a(n) for n=0..44.

MathPages, Balls In Bins With Limited Capacity

FORMULA

Refer to comment to A120987 concerning formulas for general values of r and considerations.

Therefrom we get

T(n, m) = Qsc(3, n, m) =

Nb(4*m-n, 3, n+1) = Nb(4*(n-m)+3, 3, n+1) =

Sum_{j=0..n+1} (-1)^j*Cb(n+1, j)*Cb(4*(m-j), 4*(m-j)-n) =

Sum_{j=0..m} (-1)^(m-j)*Cb(n+1, m-j)*Cb(4*j, n) =

(in this last version Cb(n,m) can be replaced by binomial(n,m))

Sum_{j=0..m} (-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n) = [z^n, t^m](1-t)/(1-t(1+(1-t)z)^4) where [x^n]F(x) denotes the coefficient of x^n in the formal power series expansion of F(x),

Nb(s,r,n) denotes the (r+1)-nomial coefficient [x^s](1+x+..+x^r)^n,(Nb(s,3,n) = A008287(n,s)).

Cb(x,m) denotes the binomial coefficient in its extended falling factorial notation (Cb(x,m)= x^_m/m! iff m is a nonnegative integer, 0 otherwise), as defined in the Graham et al. reference.

The diagonal T(n, n) = Nb(3, 3, n+1) = Sum_{j=0..n} (-1)^(n-j)*Cb(n+1, n-j)*Cb(4*j, n) = Cb(n+3, 3) = binomial(n+3, 3) = A000292(n+1).

EXAMPLE

Triangle starts:

0, 4;

0, 6, 10;

0, 4, 40, 20;

0, 1, 65, 155, 35;

0, 0, 56, 456, 456, 56;

T(3,2) = 40, which accounts for the following words:

[0 <= a <= 0, 1 | 0 <= b <= 1] = 2

[0 <= a <= 1, 2 | 0 <= b <= 2] = 6

[0 <= a <= 2, 3 | 0 <= b <= 3] = 12

[0 <= a <= 3 | 0, 1 <= b <= 3] = 12

[1 <= a <= 3 | 1, 2 <= b <= 3] = 6

[2 <= a <= 3 | 2, 3 <= b <= 3] = 2

PROG

(MuPAD) T:=(n, m)->_plus((-1)^(m-j)*binomial(n+1, m-j)*binomial(4*j, n)$j=0..m):

(PARI) T(n, k) = sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*binomial(4*j, n));

tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Feb 09 2016

CROSSREFS

Cf. A119900 (r=1, binary words), A120987 (r=2, ternary words), A008287 (quadrinomial coefficients).

Row sums give A000302.

Cf. A000292.

KEYWORD

tabl,nonn

AUTHOR

Giuliano Cabrele, Dec 13 2015

STATUS

approved

page 1 2

Search completed in 0.046 seconds