A372115 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1 + 4*x)^(1/2) * A(x) ).

1, 1, 4, 11, 48, 174, 784, 3219, 14816, 65082, 304656, 1393854, 6617184, 31086556, 149336672, 714494467, 3466785216, 16808037474, 82244904016, 402770823114, 1984987570016, 9797722907684, 48581811550112, 241324198117678, 1202874359046464, 6006605345531268

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

G.f.: A(x) = 2/(1 + sqrt(1-4*x*sqrt(1+4*x))).

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(2*k,k) * binomial(k/2,n-k)/(k+1).

D-finite with recurrence n*(n-1)*(n+1)*a(n) +2*n*(n-1)*(10*n-23)*a(n-1) +12*(n-1)*(11*n^2-64*n+83)*a(n-2) +24*(4*n^3-90*n^2+352*n-369)*a(n-3) +48*(-64*n^3+528*n^2-1433*n+1290)*a(n-4) +64*(-268*n^3+3090*n^2-11882*n+15255)*a(n-5) +192*(-208*n^3+2928*n^2-13705*n+21345)*a(n-6) -1152*(4*n-25)*(4*n-19)*(2*n-11)*a(n-7)=0. - R. J. Mathar, Apr 24 2024

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*sqrt(1+4*x))))
(PARI) a(n) = sum(k=0, n, 4^(n-k)*binomial(2*k, k)*binomial(k/2, n-k)/(k+1));

Crossrefs

Cf. A372139.

Sequence in context: A149307 A149308 A149309 * A149310 A149311 A212086

Adjacent sequences: A372112 A372113 A372114 * A372116 A372117 A372118

Keyword

nonn

Author

Seiichi Manyama, Apr 20 2024

Status

approved