A355167 -id:A355167 - cp's OEIS frontend

A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3k + 2)^n / (3^k k!).

1, 3, 12, 63, 405, 3024, 25515, 239355, 2465478, 27600669, 333051669, 4303119330, 59202612693, 863285928327, 13288589222508, 215177742933579, 3654114236490393, 64902307993517160, 1202782377224829015, 23207417212751493327, 465302639045308247262, 9677171073270491712513, 208434297638273958963225

Offset: 0

Ilya Gutkovskiy, Jun 22 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..484
Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.

Cf. A003575, A004212, A284864, A355165, A355167.

nmax = 22; CoefficientList[Series[Exp[2 x + (Exp[3 x] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] 2^(n - k) 3^k BellB[k, 1/3], {k, 0, n}], {n, 0, 22}]

E.g.f.: exp(2*x + (exp(3*x) - 1) / 3).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A004212(k).
a(n) ~ 3^(n + 2/3) * n^(n + 2/3) * exp(n/LambertW(3*n) - n - 1/3) / (sqrt(1 + LambertW(3*n)) * LambertW(3*n)^(n + 2/3)). - Vaclav Kotesovec, Jun 27 2022

A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4k + 2)^n / (4^k k!).

Original entry on oeis.org

1, 3, 13, 79, 601, 5339, 53861, 607527, 7560625, 102637235, 1506225085, 23726435583, 398852249097, 7120170905995, 134408217821205, 2673140092099543, 55832167947587425, 1221199519275467107, 27902127744298836845, 664446811342185649583, 16457968670922936733113, 423242969435491221774907

Offset: 0

Ilya Gutkovskiy, Jun 22 2022

nonn

Cf. A003576, A004213, A285064, A355164, A355167.

nmax = 21; CoefficientList[Series[Exp[2 x + (Exp[4 x] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n, k] 2^(n + k) BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(2*x + (exp(4*x) - 1) / 4).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A004213(k).
a(n) ~ 2^(2*n+1) * n^(n + 1/2) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2022

A363908 a(n) = exp(-1/5) * Sum_{k>=0} (5k + 4)^n / (5^k k!).

Original entry on oeis.org

1, 5, 30, 225, 2075, 22500, 276875, 3790625, 57050000, 934984375, 16549046875, 314146406250, 6358972578125, 136603266015625, 3101556258593750, 74164388642578125, 1861859526474609375, 48936176077929687500, 1343302192888037109375, 38425435693841064453125, 1143143051078878906250000

Offset: 0

Michael De Vlieger, Jun 27 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..450
Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.

Cf. A007405, A050488, A355164, A355167.

CoefficientList[Series[Exp[4 x + (Exp[5 x] - 1)/5], {x, 0, #}], x]* Range[0, #]! &[21]

cp's OEIS Frontend

A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3k + 2)^n / (3^k k!).

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A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4k + 2)^n / (4^k k!).

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A363908 a(n) = exp(-1/5) * Sum_{k>=0} (5k + 4)^n / (5^k k!).

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cp's OEIS Frontend

A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3*k + 2)^n / (3^k * k!).

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Formula

A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 2)^n / (4^k * k!).

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A363908 a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 4)^n / (5^k * k!).

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A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3k + 2)^n / (3^k k!).

A355165 a(n) = exp(-1/4) * Sum_{k>=0} (4k + 2)^n / (4^k k!).

A363908 a(n) = exp(-1/5) * Sum_{k>=0} (5k + 4)^n / (5^k k!).