A352859 -id:A352859 - cp's OEIS frontend

A352860 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k) * 2^k * a(k).

1, 1, 5, 67, 2273, 187411, 36539465, 16496912587, 16958655627233, 39148957534778851, 200638280176080172025, 2261092739579072893806907, 55582179517311967755693514193, 2960001149710485505367113202321491, 339497331023047752386812273780566932585

Offset: 0

Ilya Gutkovskiy, Apr 06 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..77

Cf. A000670, A122704, A126443, A352859.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] 2^k a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; A[] = 0; Do[A[x] = 1 + (Exp[x] - 1) A[2 x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] Range[0, nmax]!

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, 2^j*binomial(i, j)*v[j+1])); v; \\ Seiichi Manyama, Jun 18 2022

E.g.f. A(x) satisfies: A(x) = 1 + (exp(x) - 1) * A(2*x).

a(n) ~ c * n! * 2^(n*(n-1)/2), where c = 1.572986203588985421674040830458773854660492965929302012... - Vaclav Kotesovec, Apr 07 2022

A355109 a(n) = 1 + Sum_{k=1..n-1} binomial(n-1,k) * 2^(k-1) * a(k).

Original entry on oeis.org

1, 1, 2, 7, 44, 493, 9974, 372403, 26247008, 3559692121, 942403603562, 491777568765151, 508938530329020692, 1048381120745440503877, 4307758467916752367544414, 35349370769806113877653011083, 579693879415731511179957972407624

Offset: 0

Ilya Gutkovskiy, Jun 19 2022

nonn

Cf. A000110, A126443, A352859, A352860.

a:= proc(n) option remember; 1+add(a(k)*
      binomial(n-1, k)*2^(k-1), k=1..n-1)
    end:
seq(a(n), n=0..16);  # Alois P. Heinz, Jun 19 2022

a[n_] := a[n] = 1 + Sum[Binomial[n - 1, k] 2^(k - 1) a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; A[] = 0; Do[A[x] = (2 - x + x A[2 x/(1 - x)])/(2 (1 - x)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = (2 - x + x * A(2*x/(1 - x))) / (2 * (1 - x)).

cp's OEIS Frontend

A352860 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k) * 2^k * a(k).

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