A383254 Expansion of 1/sqrt( (1-x) * (1-5*x)^3 ).
1, 8, 51, 300, 1695, 9348, 50729, 272128, 1447155, 7643880, 40156281, 210019428, 1094338401, 5684293020, 29446107975, 152181330480, 784880109315, 4040712839880, 20768844586025, 106595697483700, 546389531720445, 2797395801163260, 14306735857573995
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..600
Programs
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Magma
[&+[(2*k+1)*Binomial(2*k, k)*Binomial(n+1, k+1): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 18 2025
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Mathematica
Table[Sum[(2*k+1)* Binomial[2*k, k]*Binomial[n+1,k+1],{k,0,n}],{n,0,28}] (* Vincenzo Librandi, May 18 2025 *) -
PARI
a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n+1, k+1));
Formula
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+1,k+1).
D-finite with recurrence n*a(n) +2*(-3*n-1)*a(n-1) +5*n*a(n-2)=0. - R. J. Mathar, May 05 2025
a(n) ~ 5^(n + 1/2) * sqrt(n/Pi). - Vaclav Kotesovec, May 05 2025
From Seiichi Manyama, Aug 19 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k) * binomial(n+1,n-k). (End)