A349715 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^4)/2 ).

1, 1, 5, 61, 1161, 30201, 998413, 40077493, 1893550865, 102951388657, 6331847746581, 434653328279853, 32944254978940825, 2732662648183661545, 246228744062320481309, 23949858491053731087781, 2501088964314980938821153, 279111248034686114681365473

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..339

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

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

E.g.f.: ( -LambertW( -2*x * exp(2*x) )/(2*x) )^(1/4).

G.f.: 2 * Sum_{k>=0} (4*k+1)^(k-1) * x^k/(2 - (4*k+1)*x)^(k+1).

a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-2) * n^(n-1) / (LambertW(exp(-1))^(n + 1/4) * exp(n)). - Vaclav Kotesovec, Nov 26 2021

Mathematica

a[n_] := (1/2^n) * Sum[(4*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (4*k+1)^(n-1)*binomial(n, k))/2^n;
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-2*x*exp(2*x))/(2*x))^(1/4)))
(PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (4*k+1)^(k-1)*x^k/(2-(4*k+1)*x)^(k+1)))

Crossrefs

Cf. A007889, A202617, A349714, A349716, A349719, A349720, A349721.

Sequence in context: A213110 A298695 A352238 * A217820 A217821 A371371

Adjacent sequences: A349712 A349713 A349714 * A349716 A349717 A349718

Keyword

nonn

Author

Seiichi Manyama, Nov 26 2021

Status

approved