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A347427 Denominators of coefficients in expansion of e.g.f. x / (1 + 2*x - exp(x)). 1

%I #8 Sep 01 2021 22:21:54

%S 1,2,6,1,30,3,42,1,90,5,22,1,2730,35,90,3,1530,45,3990,35,6930,33,

%T 2070,45,40950,91,378,7,870,15,4774,77,13090,595,210,3,383838,1729,

%U 126,21,284130,693,297990,55,217350,2415,29610,315,4873050,16575,131274,1287,157410,1485

%N Denominators of coefficients in expansion of e.g.f. x / (1 + 2*x - exp(x)).

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers</a>

%F a(n) is the denominator of b(n) where b(n) = (1/(n + 1)) * Sum_{k=1..n} binomial(n+1,k+1) * b(n-k), b(0) = 1.

%F a(n) is the denominator of b(n) where b(n) = Bernoulli(n) - 2 * Sum_{k=0..n-1} binomial(n,k) * b(k) * Bernoulli(n-k).

%e 1, 1/2, 5/6, 2, 191/30, 76/3, 5081/42, 674, 386237/90, 153704/5, 5382687/22, 2142054, 55851596621/2730, 7408761716/35, ...

%t nmax = 53; CoefficientList[Series[x/(1 + 2 x - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]! // Denominator

%t b[0] = 1; b[n_] := b[n] = (1/(n + 1)) Sum[Binomial[n + 1, k + 1] b[n - k], {k, 1, n}]; a[n_] := Denominator[b[n]]; Table[a[n], {n, 0, 53}]

%t 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_] := Denominator[b[n]]; Table[a[n], {n, 0, 53}]

%o (PARI) my(x='x+O('x^60)); apply(denominator, Vec(serlaplace(x/(1+2*x-exp(x))))) \\ _Michel Marcus_, Sep 01 2021

%Y Cf. A027641, A027642, A347426 (numerators).

%K nonn,frac

%O 0,2

%A _Ilya Gutkovskiy_, Sep 01 2021

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Last modified May 21 14:18 EDT 2024. Contains 372738 sequences. (Running on oeis4.)