A360321 a(n) = Sum_{k=0..n} 5^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
1, 2, 16, 130, 1070, 8902, 74724, 631902, 5376840, 45990070, 395106656, 3407196982, 29477061166, 255733684010, 2224098916300, 19384492018770, 169270624419390, 1480625235653670, 12970844831940000, 113785067475668550, 999400688480388570
Offset: 0
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
Table[Sum[5^(n-k) Binomial[n-1,n-k]Binomial[2k,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jun 22 2025 *) -
PARI
a(n) = sum(k=0, n, 5^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
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PARI
my(N=30, x='x+O('x^N)); Vec(sqrt((1-5*x)/(1-9*x)))
Formula
G.f.: sqrt( (1-5*x)/(1-9*x) ).
n*a(n) = 2*(7*n-6)*a(n-1) - 45*(n-2)*a(n-2).
Sum_{i=0..n} Sum_{j=0..i} (1/5)^i * a(j) * a(i-j) = (9/5)^n.
a(n) ~ 2 * 3^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 04 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)