A292893 -id:A292893 - cp's OEIS frontend

A356812 Expansion of e.g.f. exp(x * (1 - exp(2*x))).

1, 0, -4, -12, 16, 400, 2208, -448, -131840, -1357056, -4820480, 71120896, 1537308672, 14006460416, 3075702784, -2224350781440, -41354996154368, -359660395495424, 1675436608585728, 121894823709900800, 2317859245604208640, 20543311167964053504

Offset: 0

Seiichi Manyama, Aug 29 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..499

Cf. A292893, A356813.

Cf. A351736, A356815, A356819.

With[{nn=30},CoefficientList[Series[Exp[x(1-Exp[2x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2023 *)

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

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(2*k+1)*x)^(k+1)))

a(n) = sum(k=0, n, (-1)^k*(2*k+1)^(n-k)*binomial(n, k));

a(n) = n!*sum(k=0, n\2, (-1)^k*2^(n-k)*stirling(n-k, k, 2)/(n-k)!);

G.f.: Sum_{k>=0} (-x)^k / (1 - (2*k+1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (2*k+1)^(n-k) * binomial(n,k).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 2^(n-k) * Stirling2(n-k,k)/(n-k)!.

A356813 Expansion of e.g.f. exp(x * (1 - exp(3*x))).

Original entry on oeis.org

1, 0, -6, -27, 0, 1215, 12312, 45927, -657072, -15857937, -167699160, -266960529, 29356170984, 700068823623, 8419188469104, -1491045413265, -2856006296224992, -79065447339366945, -1162293393139510824, -744123842820101745, 538503788896323210360

Offset: 0

Seiichi Manyama, Aug 29 2022

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Seiichi Manyama, Table of n, a(n) for n = 0..470

Cf. A292893, A356812.

Cf. A351737, A356816.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(3*x)))))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(3*k+1)*x)^(k+1)))

a(n) = sum(k=0, n, (-1)^k*(3*k+1)^(n-k)*binomial(n, k));

a(n) = n!*sum(k=0, n\2, (-1)^k*3^(n-k)*stirling(n-k, k, 2)/(n-k)!);

G.f.: Sum_{k>=0} (-x)^k / (1 - (3*k+1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (3*k+1)^(n-k) * binomial(n,k).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 3^(n-k) * Stirling2(n-k,k)/(n-k)!.

A292894 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (1 - exp(x))).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 0, -2, 1, 1, 0, 0, -3, 1, 1, 0, 0, -6, 8, -2, 1, 0, 0, 0, -12, 55, -9, 1, 0, 0, 0, -24, -20, 84, -9, 1, 0, 0, 0, 0, -60, 330, -637, 50, 1, 0, 0, 0, 0, -120, -120, 2478, -4992, 267, 1, 0, 0, 0, 0, 0, -360, -210, 11704, -10593, 413, 1, 0, 0, 0, 0, 0, -720, -840, 19824, -15192, 92060, -2180

Offset: 0

Seiichi Manyama, Sep 26 2017

			Square array begins:
   1,  1,   1,   1,    1, ...
  -1,  0,   0,   0,    0, ...
   0, -2,   0,   0,    0, ...
   1, -3,  -6,   0,    0, ...
   1,  8, -12, -24,    0, ...
  -2, 55, -20, -60, -120, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000587, A292893, A292951.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Cf. A292892, A355607.

T(n, k) = n!*sum(j=0, n\(k+1), (-1)^j*stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} (-1)^j * Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

A356814 a(n) = Sum_{k=0..n} (-1)^k * (kn+1)^(n-k) binomial(n,k).

Original entry on oeis.org

1, 0, -4, -27, -64, 4375, 199584, 6739607, 169934848, -1012395105, -709624000000, -86599643309201, -8221227668471808, -638169258399740977, -27617164284655812608, 3853095093357099609375, 1568756883209662050074624, 360407172063462944082773311

Offset: 0

Seiichi Manyama, Aug 29 2022

sign

Cf. A292893, A356812, A356813.

Cf. A320258, A356806, A356811, A356817.

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

a(n) = n!*sum(k=0, n\2, (-1)^k*n^(n-k)*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * [x^n] exp( x * (1 - exp(n * x)) ).
a(n) = [x^n] Sum_{k>=0} (-x)^k / (1 - (n*k+1)*x)^(k+1).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * n^(n-k) * Stirling2(n-k,k)/(n-k)!.

A375683 Expansion of e.g.f. 1 / (1 + x * (exp(x) - 1)).

Original entry on oeis.org

1, 0, -2, -3, 20, 115, -306, -6307, -6616, 462663, 2956130, -38945951, -656504388, 2325876683, 145820995670, 562691968005, -33452317341616, -449954883966065, 7055017491780810, 233802046526955497, -571834988279277340, -112474674691684827501

Offset: 0

Seiichi Manyama, Aug 24 2024

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Cf. A052848, A367880, A367881.

Cf. A292893, A375684.

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

a(n) = n!*sum(k=0, n\2, (-1)^k*k!*stirling(n-k, k, 2)/(n-k)!);

a(0) = 1; a(n) = -n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * k! * Stirling2(n-k,k)/(n-k)!.

cp's OEIS Frontend

A356812 Expansion of e.g.f. exp(x * (1 - exp(2*x))).

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A356813 Expansion of e.g.f. exp(x * (1 - exp(3*x))).

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A292894 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (1 - exp(x))).

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A356814 a(n) = Sum_{k=0..n} (-1)^k * (k*n+1)^(n-k) * binomial(n,k).

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A375683 Expansion of e.g.f. 1 / (1 + x * (exp(x) - 1)).

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Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A356814 a(n) = Sum_{k=0..n} (-1)^k * (kn+1)^(n-k) binomial(n,k).