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 A349714 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^3)/2 ). 6
 1, 1, 4, 37, 532, 10426, 259300, 7823908, 277713904, 11339452792, 523621438336, 26982030104536, 1534947906550528, 95550736737542464, 6460746383585984512, 471533064029919744256, 36946948091091750496000, 3093472887944746070621056 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..352 Eric Weisstein's World of Mathematics, Lambert W-Function. FORMULA a(n) = (1/2^n) * Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k). E.g.f.: ( -LambertW( -3*x/2 * exp(3*x/2) )/(3*x/2) )^(1/3). G.f.: 2 * Sum_{k>=0} (3*k+1)^(k-1) * x^k/(2 - (3*k+1)*x)^(k+1). a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/3) * 2^n * exp(n)). - Vaclav Kotesovec, Nov 26 2021 MATHEMATICA a[n_] := (1/2^n) * Sum[(3*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, (3*k+1)^(n-1)*binomial(n, k))/2^n; (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2))^(1/3))) (PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (3*k+1)^(k-1)*x^k/(2-(3*k+1)*x)^(k+1))) CROSSREFS Cf. A007889, A202617, A349715, A349716, A349719, A349720, A349721. Sequence in context: A349254 A277638 A352237 * A121080 A001518 A185082 Adjacent sequences: A349711 A349712 A349713 * A349715 A349716 A349717 KEYWORD nonn AUTHOR Seiichi Manyama, Nov 26 2021 STATUS approved

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Last modified March 29 19:01 EDT 2023. Contains 361599 sequences. (Running on oeis4.)