A340971 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * binomial(2*k,k).
1, 3, 33, 721, 23649, 1032801, 56317969, 3682424775, 280767441537, 24456613613401, 2395993939827201, 260764460901476643, 31213273328323059169, 4075382667781540713807, 576394007453263029232497
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..322
Programs
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Mathematica
a[0] = 1; a[n_] := Sum[n^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 01 2021 *) -
PARI
a(n) = sum(k=0, n, n^k*binomial(n, k)*binomial(2*k, k));
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PARI
a(n) = polcoef(1/sqrt((1-x)*(1-(4*n+1)*x)+x*O(x^n)), n);
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PARI
a(n) = polcoef((1+(2*n+1)*x+(n*x)^2)^n, n);
Formula
a(n) = [x^n] 1/sqrt((1-x)*(1-(4*n+1)*x)).
a(n) = [x^n] (1+(2*n+1)*x+(n*x)^2)^n.
a(n) = n! * [x^n] exp((2*n+1)*x) * BesselI(0,2*n*x). - Ilya Gutkovskiy, Feb 01 2021
a(n) ~ exp(1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Nov 13 2021
From Seiichi Manyama, Aug 19 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} (4*n+1)^k * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-n)^k * (4*n+1)^(n-k) * binomial(n,k) * binomial(2*k,k). (End)