A347023 -id:A347023 - cp's OEIS frontend

A347022 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(1/5).

1, 1, 5, 50, 720, 13650, 320370, 8967720, 291538080, 10795026840, 448484788680, 20658543923280, 1044915105622800, 57572197848878400, 3432143603792520000, 220109018869587398400, 15110184224165199667200, 1105545474191480800492800, 85881534014930659599571200

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A006252, A008548, A320343, A346984, A346987, A347020, A347021, A347023.

nmax = 18; CoefficientList[Series[1/(1 - 5 Log[1 + x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A008548(k).
a(n) ~ n! * exp(1/25) / (Gamma(1/5) * 5^(1/5) * n^(4/5) * (exp(1/5) - 1)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (5 - 4*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A347020 Expansion of e.g.f. 1 / (1 - 3 * log(1 + x))^(1/3).

Original entry on oeis.org

1, 1, 3, 18, 150, 1644, 22116, 353856, 6554376, 138001896, 3254445144, 84979363248, 2433814616592, 75858381808416, 2556180134677152, 92597465283789312, 3588434497019272320, 148134619713440384640, 6489652665043455707520, 300712023388466713739520

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A006252, A007559, A320343, A335531, A346982, A347015, A347021, A347022, A347023.

nmax = 19; CoefficientList[Series[1/(1 - 3 Log[1 + x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A007559(k).
a(n) ~ n! * exp(1/9) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A347021 Expansion of e.g.f. 1 / (1 - 4 * log(1 + x))^(1/4).

Original entry on oeis.org

1, 1, 4, 32, 364, 5444, 100520, 2210760, 56406240, 1637877600, 53327583360, 1924096475520, 76198487927040, 3285955396558080, 153273199794071040, 7689131281851770880, 412809183978447306240, 23616192920003184176640, 1434201753814306170808320

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A006252, A007696, A320343, A346983, A347016, A347020, A347022, A347023.

nmax = 18; CoefficientList[Series[1/(1 - 4 Log[1 + x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A007696(k).
a(n) ~ n! * exp(1/16) / (Gamma(1/4) * 2^(1/2) * n^(3/4) * (exp(1/4) - 1)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

cp's OEIS Frontend

A347022 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(1/5).

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A347020 Expansion of e.g.f. 1 / (1 - 3 * log(1 + x))^(1/3).

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A347021 Expansion of e.g.f. 1 / (1 - 4 * log(1 + x))^(1/4).

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