A346955 -id:A346955 - cp's OEIS frontend

A346977 Expansion of e.g.f. log( 1 + (exp(x) - 1)^5 / 5! ).

1, 15, 140, 1050, 6951, 42399, 239800, 1164570, 2553551, -54771717, -1384474728, -23286667950, -339924740609, -4554547609233, -56481301888144, -630768487283886, -5665064764515849, -18095553874845909, 924820173031946752, 35413415495503624986

Offset: 5

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000481, A327506, A346955, A346974, A346975, A346976.

Cf. A346947, A346955.

nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

a(n) = Stirling2(n,5) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ -(n-1)! * 2^(1+n) * 5^n * cos(n*arctan((2*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))) / log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))) / (100*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))^2 + (5*log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))^2)^(n/2). - Vaclav Kotesovec, Aug 10 2021
a(n) = Sum_{k=1..floor(n/5)} (-1)^(k-1) * (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A346390 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^3 / 3! ).

Original entry on oeis.org

1, 6, 25, 100, 511, 3626, 30045, 262800, 2470171, 25889446, 302003065, 3821936300, 51672723831, 745789322466, 11505096936085, 189023074558600, 3288243760145491, 60319276499454686, 1164282909466221105, 23603464830964817700, 501435697062735519151

Offset: 3

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000392, A000629, A003704, A327504, A346894, A346954, A346955.

nmax = 23; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS2[n, 3] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]

my(x='x+O('x^25)); Vec(serlaplace(-log(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 09 2021

a(n) = Stirling2(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,3) * k * a(k).
a(n) ~ (n-1)! / (log(6^(1/3)+1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1736, 9030, 60355, 561550, 6188996, 69919850, 781211795, 8854058850, 106994019406, 1433756147470, 21287253921635, 339206526695750, 5630710652048216, 96341917117951890, 1708973354556320875, 31787279786739738250, 623964823224788294426

Offset: 4

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000453, A000629, A003704, A327505, A346390, A346895, A346955.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

a(n) = Stirling2(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (log(2^(3/4)*3^(1/4) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

cp's OEIS Frontend

A346977 Expansion of e.g.f. log( 1 + (exp(x) - 1)^5 / 5! ).

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A346390 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^3 / 3! ).

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A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

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