A262440 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k-1,n-k).
1, 1, 5, 22, 101, 476, 2282, 11075, 54245, 267592, 1327580, 6617128, 33110090, 166215895, 836761343, 4222640822, 21354409445, 108193910000, 549084400088, 2790744368660, 14203023709276, 72371208424880, 369170645788840, 1885051297844624
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[Binomial(n, k)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 13 2015
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Mathematica
Join[{1}, Table[Sum[ Binomial[n,k] Binomial[n+k-1, n-k], {k, n}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *) -
Maxima
a(n):=sum(binomial(n,k)*binomial(n+k-2,n-k-1),k,0,n-1)/n; A(x):=sum(a(n)*x^n,n,1,30); taylor(diff(A(x),x)/A(x)*x,x,0,10);
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PARI
a(n)=sum(k=0,n,(binomial(n,k)*binomial(n+k-1,n-k))) \\ Anders Hellström, Sep 23 2015
Formula
G.f.: x*A'(x)/A(x), where A(x) is g.f. of A109081.
Recurrence: 2*n*(2*n-1)*(38*n^3 - 210*n^2 + 377*n - 219)*a(n) = 2*(380*n^5 - 2480*n^4 + 5998*n^3 - 6598*n^2 + 3219*n - 540)*a(n-1) + 2*(n-2)*(76*n^4 - 382*n^3 + 572*n^2 - 300*n + 45)*a(n-2) + 3*(n-3)*(n-2)*(38*n^3 - 96*n^2 + 71*n - 14)*a(n-3). - Vaclav Kotesovec, Sep 23 2015
a(n) = n^2*hypergeom([1-n, 1-n, n+1], [3/2, 2], 1/4) for n >= 1. - Peter Luschny, Mar 06 2022
a(n) = [x^n] ( (1 - x + x^2) / (1 - x)^2 )^n. - Seiichi Manyama, Apr 29 2024
a(n) ~ sqrt((513 - 67*sqrt(57))^(1/3) + (513 + 67*sqrt(57))^(1/3)) * (10 + (1261 - 57*sqrt(57))^(1/3) + (1261 + 57*sqrt(57))^(1/3))^n / (19^(1/3) * sqrt(Pi*n) * 2^(n + 5/6) * 3^(n + 1/3)). - Vaclav Kotesovec, Apr 30 2024