A115990 Riordan array (1/sqrt(1-2*x-3*x^2), (1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2).

1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 19, 13, 8, 4, 1, 51, 35, 22, 12, 5, 1, 141, 96, 61, 35, 17, 6, 1, 393, 267, 171, 101, 53, 23, 7, 1, 1107, 750, 483, 291, 160, 77, 30, 8, 1, 3139, 2123, 1373, 839, 476, 244, 108, 38, 9, 1, 8953, 6046, 3923, 2423, 1406, 752, 360, 147, 47, 10

Offset: 0

Author

Paul Barry, Feb 10 2006

Keywords

• easy
• nonn
• tabl

Comments

First column is central trinomial coefficients A002426. Second column is number of directed animals of size n+1, A005773(n+1). Row sums are A005717 (number of horizontal steps in all Motzkin paths of length n). First column has e.g.f. exp(x) I_0(2x). Row sums have e.g.f. dif(exp(x) I_1(2x),x).

Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/2).

Examples

			Triangle begins
    1;
    1,  1;
    3,  2,  1;
    7,  5,  3,  1;
   19, 13,  8,  4,  1;
   51, 35, 22, 12,  5,  1;
  141, 96, 61, 35, 17,  6,  1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A115991, A005773 (k=1), A025566 (k=2), A035045 (k=3), A152948 (diag. n=k+2), .

Programs

GAP

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

Magma

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

Maple

A115990 := proc(n,k)
    add(binomial(n-k,j-k)*binomial(j,n-j),j=0..n) ;
end proc:
seq(seq(A115990(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], {j, 0, n}], {n, 0, 10}, {k, 0, n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

PARI

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

Sage

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

Formula

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