A006153 -id:A006153 - cp's OEIS frontend

A368265 Expansion of e.g.f. exp(2x) / (1 - xexp(x)).

1, 3, 12, 65, 460, 4057, 42922, 529769, 7472808, 118586033, 2090936014, 40554647377, 858082563532, 19668880007129, 485528656965762, 12841428220413593, 362276791422785488, 10859170086870710497, 344648459867067117334, 11546148650974694099201

Offset: 0

Seiichi Manyama, Dec 19 2023

Cf. A006153, A072597, A092148, A368266.

Cf. A368267, A368271.

PARI
```
a(n) = n!*sum(k=0, n, (n-k+2)^k/k!);
```

a(n) = n! * Sum_{k=0..n} (n-k+2)^k / k!.

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

A377529 Expansion of e.g.f. 1/(1 - x * exp(x))^2.

Original entry on oeis.org

1, 2, 10, 66, 560, 5770, 69852, 970886, 15228880, 266006610, 5119447700, 107617719022, 2453167135608, 60268223308826, 1587381621990556, 44619277892537910, 1333135910963656352, 42189279001183102882, 1409741875877923927332, 49597905017847180008126

Offset: 0

Seiichi Manyama, Oct 30 2024

Cf. A006153, A377530.

Cf. A377503, A377527.

With[{nn=20},CoefficientList[Series[1/(1-x Exp[x])^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 04 2025 *)

a(n) = n!*sum(k=0, n, (k+1)*k^(n-k)/(n-k)!);

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

a(n) ~ n! * n/((1 + LambertW(1))^2 * LambertW(1)^n). - Vaclav Kotesovec, Oct 31 2024

A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).

Original entry on oeis.org

1, 1, 6, 39, 352, 3965, 53556, 844123, 15204960, 308118105, 6937562980, 171826160231, 4642588564032, 135891789038629, 4283619809941668, 144674451274329075, 5211965027738046016, 199498704931954788785, 8085413817213212761668, 345895984008645703002559

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..396

Cf. A000290, A006153, A033453, A033462, A302189, A308862.

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

my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + x)*exp(x)))) \\ Michel Marcus, Mar 10 2022

E.g.f.: 1 / (1 - Sum_{k>=1} k^2*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 3*r + r^2)), where r = A201941 = 0.44413022882396659058546632949098466707932096994213775695918... is the root of the equation exp(r)*r*(1 + r) = 1. - Vaclav Kotesovec, Jun 29 2019

A346889 Expansion of e.g.f. 1 / (1 - x^3 * exp(x) / 3!).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 40, 315, 2296, 15204, 117720, 1127445, 11531740, 120909646, 1370809804, 17111895255, 227853866800, 3182209445640, 47003318806896, 737325061500009, 12187616610231540, 210930852047426770, 3821604062633503300, 72479758506840597451

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..464

Column k=3 of A351703.
Cf. A006153, A346888, A346890, A346893.
Cf. A000292, A145453, A353999.

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

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^3*exp(x)/3!))) \\ Michel Marcus, Aug 06 2021

a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(6^k*(n-3*k)!)); \\ Seiichi Manyama, May 13 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,3) * a(n-k).
a(n) ~ n! / ((1 + LambertW(2^(1/3)/3^(2/3))) * 3^(n+1) * LambertW(2^(1/3)/3^(2/3))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-3*k)/(6^k * (n-3*k)!). - Seiichi Manyama, May 13 2022

A358080 Expansion of e.g.f. 1/(1 - x^2 * exp(x)).

Original entry on oeis.org

1, 0, 2, 6, 36, 260, 2190, 21882, 248696, 3181320, 45229050, 707208590, 12063902532, 222939837276, 4436813677478, 94605994108290, 2151763873634160, 51999544476324752, 1330540380342907506, 35936656483848501654, 1021700660649312689660

Offset: 0

Seiichi Manyama, Oct 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..425

Cf. A006153, A358081.

Cf. A216507, A345747, A358064.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*exp(x))))

a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/(n - 2*k)!.

a(n) ~ n! / ((1 + LambertW(1/2)) * 2^(n+1) * LambertW(1/2)^n). - Vaclav Kotesovec, Oct 30 2022

A092148 Expansion of e.g.f. 1/(exp(x)-xexp(2x)).

Original entry on oeis.org

1, 0, 3, 11, 85, 739, 7831, 96641, 1363209, 21632759, 381433771, 7398080029, 156533563693, 3588046200179, 88571349871551, 2342565398442569, 66087436823953681, 1980956920420309231, 62871632567144951635, 2106277265332074827573, 74276723394195659799861

Offset: 0

Ralf Stephan, Mar 31 2004

nonn

Cf. A006153, A072597.

With[{nn=20},CoefficientList[Series[1/(Exp[x]-x Exp[2x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 19 2020 *)

PARI
```
a(n)=n!*sum(k=0,n,(n-k-1)^k/k!)
```

a(n) = n! * Sum_{k=0..n} (n-k-1)^k/k!. [Corrected by Georg Fischer, Jun 22 2022]

a(n) ~ n! / ((LambertW(1) + 1) * LambertW(1)^(n-1)). - Vaclav Kotesovec, Jun 22 2022

Corrected and extended by Harvey P. Dale, Sep 19 2020

A308862 Expansion of e.g.f. 1/(1 - x(1 + 3x + x^2)*exp(x)).

Original entry on oeis.org

1, 1, 10, 81, 976, 14505, 258456, 5377897, 127852096, 3419620209, 101625743080, 3322169384721, 118475520287136, 4577175039397753, 190436902905933880, 8489222610046324665, 403657900923994965376, 20393319895130130117729, 1090902632352025316904648

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381

Cf. A000578, A006153, A144109, A279358, A302870, A308861.

nmax = 18; CoefficientList[Series[1/(1 - x (1 + 3 x + x^2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + 3*x + x^2)*exp(x)))) \\ Michel Marcus, Mar 10 2022

E.g.f.: 1 / (1 - Sum_{k>=1} k^3*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).
a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 7*r + 6*r^2 + r^3)), where r = 0.33649177041401456061485914122406146158245451810028937972189... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - Vaclav Kotesovec, Jun 29 2019

A346888 Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).

Original entry on oeis.org

1, 0, 1, 3, 12, 70, 465, 3591, 31948, 319068, 3539385, 43205635, 575312826, 8298867798, 128921967265, 2145837600375, 38097353658120, 718657756980376, 14354000800751313, 302625047150614179, 6716038666999745710, 156498725047355717250, 3820426102008414736761

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..442

Column k=2 of A351703.
Cf. A006153, A346889, A346890, A346893.
Cf. A000217, A133189, A308946, A353998.

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

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^2*exp(x)/2))) \\ Michel Marcus, Aug 06 2021

a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(2^k*(n-2*k)!)); \\ Seiichi Manyama, May 13 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,2) * a(n-k).
a(n) ~ n! / ((1 + LambertW(1/sqrt(2))) * 2^(n+1) * LambertW(1/sqrt(2))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k)/(2^k * (n-2*k)!). - Seiichi Manyama, May 13 2022

A346890 Expansion of e.g.f. 1 / (1 - x^4 * exp(x) / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 15, 35, 140, 1386, 12810, 92730, 589545, 4234945, 41832791, 483334215, 5401798220, 57262207380, 626438655900, 7740130412796, 107197808258745, 1546730804858085, 22360919412385015, 329241486278715395, 5121840342205301946

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..482

Column k=4 of A351703.
Cf. A006153, A346888, A346889, A346893.
Cf. A000332, A145454.

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

my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^4*exp(x)/4!))) \\ Michel Marcus, Aug 06 2021

a(n) = n!*sum(k=0, n\4, k^(n-4*k)/(24^k*(n-4*k)!)); \\ Seiichi Manyama, May 13 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * binomial(k,4) * a(n-k).
a(n) ~ n! / ((1 + LambertW(3^(1/4)/2^(5/4))) * 4^(n + 1) * LambertW(3^(1/4)/2^(5/4))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = n! * Sum_{k=0..floor(n/4)} k^(n-4*k)/(24^k * (n-4*k)!). - Seiichi Manyama, May 13 2022

A351703 Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 4, 1, 0, 1, 21, 1, 0, 0, 3, 148, 1, 0, 0, 1, 12, 1305, 1, 0, 0, 0, 4, 70, 13806, 1, 0, 0, 0, 1, 10, 465, 170401, 1, 0, 0, 0, 0, 5, 40, 3591, 2403640, 1, 0, 0, 0, 0, 1, 15, 315, 31948, 38143377, 1, 0, 0, 0, 0, 0, 6, 35, 2296, 319068, 672552730

Offset: 0

Seiichi Manyama, Feb 20 2022

			Square array begins:
      1,   1,  1,  1, 1, 1, ...
      1,   0,  0,  0, 0, 0, ...
      4,   1,  0,  0, 0, 0, ...
     21,   3,  1,  0, 0, 0, ...
    148,  12,  4,  1, 0, 0, ...
   1305,  70, 10,  5, 1, 0, ...
  13806, 465, 40, 15, 6, 1, ...

Column k=1..5 gives A006153, A346888, A346889, A346890, A346893.

Cf. A143398, A351761.

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

T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

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

T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(k!^j * (n-k*j)!). - Seiichi Manyama, May 13 2022

cp's OEIS Frontend

A368265 Expansion of e.g.f. exp(2*x) / (1 - x*exp(x)).

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A377529 Expansion of e.g.f. 1/(1 - x * exp(x))^2.

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A308861 Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).

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A346889 Expansion of e.g.f. 1 / (1 - x^3 * exp(x) / 3!).

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A358080 Expansion of e.g.f. 1/(1 - x^2 * exp(x)).

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A092148 Expansion of e.g.f. 1/(exp(x)-x*exp(2*x)).

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A308862 Expansion of e.g.f. 1/(1 - x*(1 + 3*x + x^2)*exp(x)).

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A346888 Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).

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A368265 Expansion of e.g.f. exp(2x) / (1 - xexp(x)).

A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).

A092148 Expansion of e.g.f. 1/(exp(x)-xexp(2x)).

A308862 Expansion of e.g.f. 1/(1 - x(1 + 3x + x^2)*exp(x)).