A114496 a(n) = Sum of binomial(n,k)*binomial(2n+k,k) over all k.

1, 4, 26, 190, 1462, 11584, 93536, 765314, 6323270, 52638760, 440815036, 3709445084, 31340292076, 265683004240, 2258793820988, 19251776923210, 164440378882630, 1407266585304760, 12063701803046300, 103571977632247076

Offset: 0

Author

Eric Rowland, Dec 01 2005

Keywords

• nonn
• easy

Comments

Modification of A001850 inspired by the Apéry numbers A005259.

From Paul Barry, Feb 17 2009: (Start)

Central coefficient of (1 + 4x + 5x^2 + 2x^3)^n. The coefficients are the 4th row of A029635.

The third row of A029635 corresponds to the central Delannoy numbers A001850. (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Crossrefs

Cf. A114497, A114498, A003168, A156894.

Cf. A156886. - Paul Barry, Feb 17 2009

Programs

Mathematica

Table[Sum[Binomial[n, k]*Binomial[2n+k, k], {k, 0, n}], {n,0,25}]

PARI

a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n,k)*binomial(n,k));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 05 2015

Formula

a(n) = Sum_{k=0..n} (binomial(n,k)*binomial(2n+k,k)).

Recurrence: 20*n*(2*n - 1)*a(n) = (371*n^2 - 411*n + 120)*a(n-1) -2*(81*n^2 - 299*n + 278)*a(n-2) + 4*(n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ sqrt(1734 + 442*sqrt(17))*((71 + 17*sqrt(17))/16)^n/(68*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012

From Peter Bala, Oct 05 2015: (Start)

a(n) = Sum_{i = 0..n} 2^(n-i)*binomial(2*n,i)*binomial(n,i).

4*n*(2*n - 1)*(17*n - 23)*a(n) = (1207*n^3 - 2840*n^2 + 1897*n - 360)*a(n-1) - 2*(n - 1)*(17*n - 6)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 4.

1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + ... is the o.g.f. for A003168. (End)