OFFSET
0,4
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).
LINKS
G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
FORMULA
Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j).
EXAMPLE
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;
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 *)
PROG
(PARI) {T(n, k) = sum(j=0, n, 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
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Feb 10 2006
STATUS
approved