A235328 -id:A235328 - cp's OEIS frontend

A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

1, 1, 6, 42, 392, 4600, 64752, 1063216, 19952256, 421227648, 9880951040, 254960721664, 7176891675648, 218857588139008, 7187394935347200, 252897556424140800, 9491754142468702208, 378509920569294684160, 15982018774576565649408, 712306819507400060502016

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..394

Column k=2 of A351790.

Cf. A001787, A006153, A122704, A216794, A235328, A336947, A336951, A336952.

nmax = 19; CoefficientList[Series[1/(1 - x Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(2 (n - k))^k/k!, {k, 0, n}], {n, 1, 19}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

seq(n)={ Vec(serlaplace(1 / (1 - x*exp(2*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (2 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 2^(k-1) * a(n-k).
a(n) ~ n! * (2/LambertW(2))^n / (1 + LambertW(2)). - Vaclav Kotesovec, Aug 09 2021

A336951 E.g.f.: 1 / (1 - x * exp(3*x)).

Original entry on oeis.org

1, 1, 8, 69, 780, 11145, 191178, 3823785, 87406056, 2247785073, 64228084110, 2018771719569, 69221032558956, 2571290056399545, 102860527370221026, 4408690840306136505, 201557641172689004112, 9790792086366911655009, 503570143277542340304534

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..383

Column k=3 of A351790.

Cf. A006153, A027471, A235328, A255927, A328182, A336948, A336950, A336952.

nmax = 18; CoefficientList[Series[1/(1 - x Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(3 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

seq(n)={ Vec(serlaplace(1 / (1 - x*exp(3*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (3 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 3^(k-1) * a(n-k).
a(n) ~ n! * (3/LambertW(3))^n / (1 + LambertW(3)). - Vaclav Kotesovec, Aug 09 2021

A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

Original entry on oeis.org

1, 1, 10, 102, 1336, 22200, 443664, 10334128, 275060608, 8236914048, 274069953280, 10031110907136, 400520747437056, 17324601073921024, 807023462798608384, 40278407730378332160, 2144307919689898491904, 121291661335680615284736, 7264376142168665821741056

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..375

Column k=4 of A351790.

Cf. A002697, A006153, A235328, A326324, A328183, A336950, A336951.

nmax = 18; CoefficientList[Series[1/(1 - x Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(4 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

seq(n)={ Vec(serlaplace(1 / (1 - x*exp(4*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (4 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 4^(k-1) * a(n-k).
a(n) ~ n! * (4/LambertW(4))^n / (1 + LambertW(4)). - Vaclav Kotesovec, Aug 09 2021

A351790 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (k * (n-j))^j/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 6, 21, 24, 1, 1, 8, 42, 148, 120, 1, 1, 10, 69, 392, 1305, 720, 1, 1, 12, 102, 780, 4600, 13806, 5040, 1, 1, 14, 141, 1336, 11145, 64752, 170401, 40320, 1, 1, 16, 186, 2084, 22200, 191178, 1063216, 2403640, 362880

Offset: 0

Seiichi Manyama, Feb 19 2022

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,     1,     1,     1, ...
    2,    4,    6,     8,    10,    12, ...
    6,   21,   42,    69,   102,   141, ...
   24,  148,  392,   780,  1336,  2084, ...
  120, 1305, 4600, 11145, 22200, 39145, ...

Columns k=0..4 give A000142, A006153, A336950, A336951, A336952.
Main diagonal gives A235328.
Cf. A351761, A351791.

T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)

T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);

T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));

E.g.f. of column k: 1/(1 - x*exp(k*x)).

T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} k^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.

A336949 a(n) = n! * [x^n] 1 / (exp(-n*x) - x).

Original entry on oeis.org

1, 2, 14, 195, 4440, 147745, 6698448, 394852577, 29250137472, 2652483234033, 288363456748800, 36952298766628465, 5504130616452258816, 941845623036360908489, 183298110723156455921664, 40221612394630225987208625, 9876429434585097671993032704

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Cf. A063170, A072597, A235328, A336947, A336948.

Table[n! SeriesCoefficient[1/(Exp[-n x] - x), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! Sum[(n (n - k + 1))^k/k!, {k, 0, n}], {n, 1, 16}]]

a(n)={n!*polcoef(1/(exp(-n*x + O(x*x^n)) - x), n)} \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (n * (n-k+1))^k / k!.

A351795 a(n) = n! * Sum_{k=0..n} (k * (n-k))^k/k!.

Original entry on oeis.org

1, 1, 4, 30, 396, 8360, 256470, 10619952, 564959528, 37370475648, 3001942868490, 287388158562560, 32278318416029532, 4197544986996581376, 625014083479647028622, 105554855135062180485120, 20053957030647088382195280, 4255329207190209023134564352

Offset: 0

Seiichi Manyama, Feb 19 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..266
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A235328, A349965, A351768, A351796.

a[n_] := n!*(1 + Sum[(k*(n - k))^k/k!, {k, 1, n}]); Array[a, 18, 0] (* Amiram Eldar, Feb 19 2022 *)

PARI
```
a(n) = n!*sum(k=0, n, (k*(n-k))^k/k!);
```

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / (sqrt(LambertW(exp(2)*n)^2 - 1) * exp(n*(1 - 1/LambertW(exp(2)*n))) * LambertW(exp(2)*n)^n). - Vaclav Kotesovec, Feb 20 2022

A302398 a(n) = n! * [x^n] 1/(1 + xexp(nx)).

Original entry on oeis.org

1, -1, -2, 3, 248, 5655, 62064, -3516625, -376936064, -21890186577, -495165203200, 96687112380639, 20607024735783936, 2471270260977141767, 142697263160045590528, -25986252776953159328625, -11860424645318274482077696, -2719428501410438623907546529, -372732332273232481973818294272

Offset: 0

Ilya Gutkovskiy, Apr 07 2018

sign

Cf. A134095, A235328, A302397.

Table[n! SeriesCoefficient[1/(1 + x Exp[n x]), {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! Sum[(-1)^(n - k) (n (n - k))^k/k!, {k, 0, n}], {n, 18}]]
Join[{1}, Table[Sum[(-1)^k k! (n k)^(n - k) Binomial[n, k], {k, 0, n}], {n, 18}]]

a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*(n*(n-k))^k/k!.

a(n) = Sum_{k=0..n} (-1)^k*k!*(n*k)^(n-k)*binomial(n,k).

cp's OEIS Frontend

A336950 E.g.f.: 1 / (1 - x * exp(2*x)).

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A336951 E.g.f.: 1 / (1 - x * exp(3*x)).

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A336952 E.g.f.: 1 / (1 - x * exp(4*x)).

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A351790 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (k * (n-j))^j/j!.

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A336949 a(n) = n! * [x^n] 1 / (exp(-n*x) - x).

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A351795 a(n) = n! * Sum_{k=0..n} (k * (n-k))^k/k!.

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A302398 a(n) = n! * [x^n] 1/(1 + x*exp(n*x)).

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Mathematica

Formula

A302398 a(n) = n! * [x^n] 1/(1 + xexp(nx)).