A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j)*2^(n-j).

1, 1, 1, 5, 3, 1, 13, 9, 5, 1, 49, 31, 17, 7, 1, 161, 105, 61, 29, 9, 1, 581, 371, 217, 111, 45, 11, 1, 2045, 1313, 781, 417, 189, 65, 13, 1, 7393, 4719, 2825, 1551, 753, 303, 89, 15, 1, 26689, 17041, 10277, 5757, 2921, 1289, 461, 117, 17, 1

Offset: 0

Author

Paul Barry, Feb 10 2006

Keywords

• easy
• nonn
• tabl

Comments

First column is A084601 with e.g.f. exp(x) Bessel_I(0,2*sqrt(2)x). Row sums are A098518(n+1) with e.g.f. dif(exp(x) Bessel_I(1,2*sqrt(2)x)/sqrt(2)).

Riordan array (1/sqrt(1-2*x-7*x^2), (1+x-sqrt(1-2*x-7*x^2))/2).

Examples

			Triangle begins as:
    1;
    1,   1;
    5,   3,   1;
   13,   9,   5,   1;
   49,  31,  17,   7,  1;
  161, 105,  61,  29,  9,  1;
  581, 371, 217, 111, 45, 11, 1;
		

Links

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

Crossrefs

Cf. A084601 (k=0), A098518.

Programs

GAP

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j)) ))); # G. C. Greubel, May 09 2019

Magma

[[(&+[Binomial(n-k, j-k)*Binomial(j, n-j)*2^(n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

Maple

A115991 := proc(n,k)
    add(binomial(n-k,j-k)*binomial(j,n-j)*2^(n-j),j=0..n) ;
end proc:
seq(seq(A115991(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 25 2023

Mathematica

Table[Sum[Binomial[n-k, j-k]*Binomial[j, n-j]*2^(n-j), {j, 0, n}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 09 2019 *)

PARI

{T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j))}; \\ G. C. Greubel, May 09 2019

Sage

[[sum(binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019