A346754 -id:A346754 - cp's OEIS frontend

A346751 Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

0, 0, 0, 1, 4, 10, 10, -105, -1064, -6076, -16680, 129525, 2642860, 25431406, 130210444, -639438345, -26431524560, -382074099000, -3083015556624, 5641134587049, 726952330301940, 14940678486798610, 173111303303845060, 258953439321230731, -43858702741534022936

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..471

Cf. A000292, A009306, A145453, A346750, A346752, A346754.

nmax = 24; CoefficientList[Series[Log[1 + x^3 Exp[x]/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 3] - (1/n) Sum[Binomial[n, k] Binomial[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 24}]

a(0) = 0; a(n) = binomial(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,3) * k * a(k).

a(n) = n! * Sum_{k=1..floor(n/3)} (-1)^(k-1) * k^(n-3*k-1)/(6^k * (n-3*k)!). - Seiichi Manyama, Dec 14 2023

A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

Original entry on oeis.org

0, 0, 1, 3, 9, 40, 225, 1491, 11578, 102852, 1026945, 11394955, 139091106, 1852061718, 26716291693, 415033647315, 6908006807640, 122645325067576, 2313546734841633, 46209268921868595, 974228913850588750, 21620679147700290210, 503810188866302511501

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

nonn

Cf. A000217, A009444, A133189, A292953, A305990, A346750, A346754, A346755.

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

a(0) = 0; a(n) = binomial(n,2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,2) * k * a(k).
a(n) ~ (n-1)! / (2*LambertW(1/sqrt(2)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023

A346755 Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 105, 756, 6510, 46530, 289245, 1892605, 16187171, 170721915, 1833783770, 18875258780, 196470797580, 2255939795436, 29179692064545, 401813199660285, 5612352516200815, 79620308330422475, 1182881543312932386

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

nonn

Cf. A000332, A009444, A145454, A292955, A305990, A346752, A346753, A346754.

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

a(0) = 0; a(n) = binomial(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (4*LambertW(3^(1/4)/2^(5/4)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/4)} k^(n-4*k-1)/(24^k * (n-4*k)!). - Seiichi Manyama, Dec 14 2023

A366546 Expansion of e.g.f. -log(1 - x^3 * exp(x)).

Original entry on oeis.org

0, 0, 0, 6, 24, 60, 480, 5250, 40656, 363384, 4839120, 65198430, 859543080, 13311494196, 233478687624, 4190929145130, 79746180437280, 1667320408619760, 36965002127643936, 854734007793179574, 20962277675893792440, 544839141515795731500

Offset: 0

Seiichi Manyama, Dec 14 2023

nonn

Cf. A009444, A366459.

Cf. A292889, A346754, A358081.

a(n) = n!*sum(k=1, n\3, k^(n-3*k-1)/(n-3*k)!);

a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(n-3*k)!.

cp's OEIS Frontend

A346751 Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

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A346753 Expansion of e.g.f. -log( 1 - x^2 * exp(x) / 2 ).

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A346755 Expansion of e.g.f. -log( 1 - x^4 * exp(x) / 4! ).

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A366546 Expansion of e.g.f. -log(1 - x^3 * exp(x)).

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