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 A347426 Numerators of coefficients in expansion of e.g.f. x / (1 + 2*x - exp(x)). 1
 1, 1, 5, 2, 191, 76, 5081, 674, 386237, 153704, 5382687, 2142054, 55851596621, 7408761716, 212280163009, 84477431614, 548645908536349, 218335036876496, 277343253225866383, 36789818034038954, 115953566271323978161, 9228803413607663876, 10136447282859468915079 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..22. Index entries for sequences related to Bernoulli numbers FORMULA a(n) is the numerator of b(n) where b(n) = (1/(n + 1)) * Sum_{k=1..n} binomial(n+1,k+1) * b(n-k), b(0) = 1. a(n) is the numerator of b(n) where b(n) = Bernoulli(n) - 2 * Sum_{k=0..n-1} binomial(n,k) * b(k) * Bernoulli(n-k). EXAMPLE 1, 1/2, 5/6, 2, 191/30, 76/3, 5081/42, 674, 386237/90, 153704/5, 5382687/22, 2142054, 55851596621/2730, 7408761716/35, ... MATHEMATICA nmax = 22; CoefficientList[Series[x/(1 + 2 x - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]! // Numerator b[0] = 1; b[n_] := b[n] = (1/(n + 1)) Sum[Binomial[n + 1, k + 1] b[n - k], {k, 1, n}]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 22}] b[0] = 1; b[n_] := b[n] = BernoulliB[n] - 2 Sum[Binomial[n, k] b[k] BernoulliB[n - k], {k, 0, n - 1}]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 22}] PROG (PARI) my(x='x+O('x^30)); apply(numerator, Vec(serlaplace(x/(1+2*x-exp(x))))) \\ Michel Marcus, Sep 01 2021 CROSSREFS Cf. A027641, A027642, A347427 (denominators). Sequence in context: A142599 A266461 A068566 * A212481 A046270 A139207 Adjacent sequences: A347423 A347424 A347425 * A347427 A347428 A347429 KEYWORD nonn,frac AUTHOR Ilya Gutkovskiy, Sep 01 2021 STATUS approved

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Last modified April 19 23:40 EDT 2024. Contains 371798 sequences. (Running on oeis4.)