A346753 -id:A346753 - cp's OEIS frontend

A346750 Expansion of e.g.f. log( 1 + x^2 * exp(x) / 2 ).

0, 0, 1, 3, 3, -20, -135, -189, 3598, 33300, 39105, -2164085, -23831214, -5268042, 3038813869, 36984819795, -59749871880, -8207734934984, -105142191601887, 482549202944307, 37754304692254030, 489494512692093090, -4466445363328684659, -271973408844483808517

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..457

Cf. A000217, A009306, A133189, A300455, A346751, A346752, A346753.

nmax = 23; 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, 23}]

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! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * k^(n-2*k-1)/(2^k * (n-2*k)!). - Seiichi Manyama, Dec 14 2023

A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 30, 175, 1176, 7364, 50520, 425205, 4010380, 39433966, 414654604, 4793188855, 59834495280, 789420239560, 11016095913456, 163423065359529, 2565467553034740, 42320595474149650, 732058678770177220, 13275485607004016011

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

nonn

Cf. A000292, A009444, A145453, A292954, A305990, A346751, A346753, A346755.

nmax = 23; 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, 23}]

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-1)! / (3*LambertW(2^(1/3)/3^(2/3)))^n. - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(6^k * (n-3*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

A366459 Expansion of e.g.f. -log(1 - x^2 * exp(x)).

Original entry on oeis.org

0, 0, 2, 6, 24, 140, 990, 8442, 84056, 955656, 12227130, 173812430, 2717859012, 46362339036, 856770362630, 17050946225250, 363576478312560, 8269357341437072, 199837364514425586, 5113346326011170838, 138106722548779770620, 3926456810081828991780

Offset: 0

Seiichi Manyama, Dec 14 2023

nonn

Cf. A009444, A366546.

Cf. A216507, A346753, A358080.

a(n) = n!*sum(k=1, n\2, k^(n-2*k-1)/(n-2*k)!);

a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(n-2*k)!.

a(n) ~ (n-1)! / (2^n *LambertW(1/2)^n). - Vaclav Kotesovec, Dec 29 2023

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A346750 Expansion of e.g.f. log( 1 + x^2 * exp(x) / 2 ).

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A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

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

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A366459 Expansion of e.g.f. -log(1 - x^2 * exp(x)).

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