A114422 Riordan array (1/sqrt(1-2*x-3*x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.

1, 1, 1, 3, 3, 1, 7, 9, 5, 1, 19, 26, 19, 7, 1, 51, 75, 65, 33, 9, 1, 141, 216, 211, 132, 51, 11, 1, 393, 623, 665, 483, 235, 73, 13, 1, 1107, 1800, 2058, 1674, 963, 382, 99, 15, 1, 3139, 5211, 6294, 5598, 3663, 1739, 581, 129, 17, 1, 8953, 15115, 19095, 18261, 13243

Offset

0,4

Comments

First column is central trinomial numbers A002426.

Second column is A005774.

Third column is A025568.

Row sums are A116387.

Diagonal sums are A116388.

Product of A007318 and A116382.

Column k has e.g.f. exp(x)*Sum_{j=0..k} C(k,j)*Bessel_I(k+j,2*x).

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

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

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

Example

Triangle begins
1,
1, 1,
3, 3, 1,
7, 9, 5, 1,
19, 26, 19, 7, 1,
51, 75, 65, 33, 9, 1,
141, 216, 211, 132, 51, 11, 1

Mathematica

T[n_, k_] := Sum[Binomial[n, j - k]*Binomial[j, n - j], {j, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

Prog

(PARI) {T(n, k) = sum(j=0, n, binomial(n, j-k)*binomial(j, n-j))};
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 15 2018
(Magma) [[(&+[Binomial(n, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Dec 15 2018
(Sage) [[sum(binomial(n, j-k)*binomial(j, n-j) for j in range(n+1)) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Dec 15 2018
(GAP) T:=Flat(List([0..10], n->List([0..n], k->Sum([0..n], j-> Binomial(n, j-k)*Binomial(j, n-j))))); # G. C. Greubel, Dec 15 2018

Crossrefs

Sequence in context: A344726 A116401 A106479 * A127501 A181304 A339435

Adjacent sequences: A114419 A114420 A114421 * A114423 A114424 A114425

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 12 2006

Status

approved