A154948 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A154948

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

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 3, 10, 14, 7, 1, 3, 15, 30, 26, 9, 1, 4, 21, 55, 70, 42, 11, 1, 4, 28, 91, 155, 138, 62, 13, 1, 5, 36, 140, 301, 363, 242, 86, 15, 1, 5, 45, 204, 532, 819, 743, 390, 114, 17, 1, 6, 55, 285, 876, 1652, 1925, 1375, 590, 146, 19, 1

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

OFFSET

0,4

COMMENTS

Row sums are A113300(n+1). Diagonal sums are A154949.

Product of A154950 and A007318.

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

Number triangle T(n,k) = Sum_{j=0..n+1} C(n+1-j,k+1)*C(k-1,j).

T(n, k) = binomial(n+1,k+1)*2F1(-(n-k), -(k-1); -(n+1); -1). - G. C. Greubel, Feb 18 2020

EXAMPLE

Triangle begins

1, 1;

2, 3, 1;

2, 6, 5, 1;

3, 10, 14, 7, 1;

3, 15, 30, 26, 9, 1;

4, 21, 55, 70, 42, 11, 1;

MAPLE

seq(seq( add(binomial(k-1, j)*binomial(n-j+1, k+1), j=0..n+1), k=0..n), n=0..10); # G. C. Greubel, Feb 18 2020

MATHEMATICA

Table[Binomial[n+1, k+1]*Hypergeometric2F1[-n+k, -k+1, -n-1, -1], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

PROG

(Magma) [ (&+[Binomial(k-1, j)*Binomial(n-j+1, k+1): j in [0..n+1]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

(Sage) [[ sum(binomial(k-1, j)*binomial(n-j+1, k+1) for j in (0..n+1)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 18 2020

CROSSREFS

Sequence in context: A181176 A131108 A128255 * A109091 A138507 A209579

Adjacent sequences: A154945 A154946 A154947 * A154949 A154950 A154951

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Jan 17 2009

EXTENSIONS

a(45)=0 removed by Georg Fischer, Feb 18 2020

STATUS

approved