A337556 -id:A337556 - cp's OEIS frontend

A337555 a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n,k) * (3^k-1) * a(n-k).

1, 1, 6, 43, 408, 4861, 69516, 1159663, 22108848, 474192601, 11300589876, 296237533483, 8471642214888, 262456441714741, 8756520140416236, 313017838828154503, 11935355244756882528, 483537933291091103281, 20741938090482567562596, 939180816648348685174723

Offset: 0

Ilya Gutkovskiy, Aug 31 2020

nonn

Cf. A000556, A003462, A337556.

a[0] = 1; a[n_] := a[n] = (1/2) Sum[Binomial[n, k] (3^k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[2/(2 + Exp[x] - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(2 / (2 + exp(x + O(x*x^n)) - exp(3*x + O(x*x^n)))))} \\ Andrew Howroyd, Aug 31 2020

E.g.f.: 2 / (2 + exp(x) - exp(3*x)).

a(n) ~ n! / ((r+3) * log(r)^(n+1)), where r = 1.52137970680456756960408... is the real root of the equation r^3 - r = 2. - Vaclav Kotesovec, Aug 31 2020

A376113 a(0) = 1; a(n) = (1/3) * Sum_{k=1..n} (4^k-1) * a(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 6, 137, 11938, 4095882, 5599192492, 30588428274345, 668265444025582410, 58395039572032120897838, 20410643002515607839683651348, 28536181214271796693200339702494058, 159585939576145805663910944364491926768148, 3569877304419418296304606194938539586766279745396

Offset: 0

Ilya Gutkovskiy, Sep 10 2024

nonn

Cf. A015085, A337556, A376111, A376112.

a[0] = 1; a[n_] := a[n] = (1/3) Sum[(4^k - 1) a[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 13}]
nmax = 13; A[] = 0; Do[A[x] = 3/(3 + x A[x] - 4 x A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 3 / (3 + x * A(x) - 4 * x * A(4*x)).

cp's OEIS Frontend

A337555 a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} binomial(n,k) * (3^k-1) * a(n-k).

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