A337011 -id:A337011 - cp's OEIS frontend

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2k + 3)^n / (2^k k!).

1, 4, 18, 92, 532, 3440, 24552, 191280, 1612304, 14597952, 141123872, 1449324992, 15743376704, 180203389696, 2166381979264, 27274611880704, 358690234163456, 4916123783848960, 70076765972288000, 1036967662211324928, 15902394743591408640

Offset: 0

Ilya Gutkovskiy, Aug 11 2020

nonn

Cf. A004211, A005494, A007405, A337011.

nmax = 20; CoefficientList[Series[Exp[3 x + (Exp[2 x] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k - 1) a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: exp(3*x + (exp(2*x) - 1) / 2).
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=2..n} binomial(n-1,k-1) * 2^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k+1).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * A004211(k).

A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2k + n)^n / (2^k k!).

Original entry on oeis.org

1, 2, 11, 92, 1025, 14232, 236403, 4568720, 100670529, 2490511776, 68341981051, 2059882505408, 67645498798721, 2403948686290816, 91914992104815459, 3762299973887526144, 164148252324092964993, 7604537914425558921728, 372812121514187124192875

Offset: 0

Ilya Gutkovskiy, Aug 11 2020

nonn

Cf. A004211, A007405, A134980, A337010, A337011.

Table[n! SeriesCoefficient[Exp[n x + (Exp[2 x] - 1)/2], {x, 0, n}], {n, 0, 18}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] n^(n - k) 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 18}]

a(n) = n! * [x^n] exp(n*x + (exp(2*x) - 1) / 2).

a(n) = Sum_{k=0..n} binomial(n,k) * n^(n-k) * A004211(k).

cp's OEIS Frontend

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2k + 3)^n / (2^k k!).

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Formula

A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2k + n)^n / (2^k k!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 3)^n / (2^k * k!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + n)^n / (2^k * k!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A337010 a(n) = exp(-1/2) * Sum_{k>=0} (2k + 3)^n / (2^k k!).

A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2k + n)^n / (2^k k!).