A357703 Expansion of e.g.f. cosh( sqrt(3) * log(1-x) ).

1, 0, 3, 9, 42, 240, 1614, 12474, 108900, 1059696, 11371932, 133410420, 1698541416, 23324023008, 343606235544, 5405580540360, 90445832210448, 1603781918563968, 30042007763367600, 592788643008571152, 12289695299276133024, 267079782474700715520

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..449

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

a(n) = Sum_{k=0..floor(n/2)} 3^k * |Stirling1(n,2*k)|.

a(n) = ( (sqrt(3))_n + (-sqrt(3))_n )/2, where (x)_n is the Pochhammer symbol.

a(0) = 1, a(1) = 0; a(n) = (2*n-3) * a(n-1) - (n^2-4*n+1) * a(n-2).

Prog

(PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cosh(sqrt(3)*log(1-x)))))
(PARI) a(n) = sum(k=0, n\2, 3^k*abs(stirling(n, 2*k, 1)));
(PARI) a(n) = round((prod(k=0, n-1, sqrt(3)+k)+prod(k=0, n-1, -sqrt(3)+k)))/2;
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; v[2]=0; for(i=2, n, v[i+1]=(2*i-3)*v[i]-(i^2-4*i+1)*v[i-1]); v;

Crossrefs

Column k=3 of A357712.

Cf. A357615.

Sequence in context: A020128 A018995 A018965 * A105804 A009402 A009584

Adjacent sequences: A357700 A357701 A357702 * A357704 A357705 A357706

Keyword

nonn

Author

Seiichi Manyama, Oct 10 2022

Status

approved