A144109 -id:A144109 - cp's OEIS frontend

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

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

A301655 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k).

Original entry on oeis.org

1, 1, 5, 44, 723, 24655, 1715816, 239697569, 69557364821, 41297123651644, 49900451628509015, 125141540794392423599, 641579398300246011553552, 6729809577032172543373047313, 146355880526667013027682326650073, 6505380999057202235872595196799580684

Offset: 0

Ilya Gutkovskiy, Mar 25 2018

nonn

Number of compositions (ordered partitions) of n where there are k^n sorts of part k.

a(n) is the n-th term of invert transform of n-th powers.

Vaclav Kotesovec, Table of n, a(n) for n = 0..79
N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Main diagonal of A320251.

Cf. A001906, A033453, A053506, A144109, A193678, A221460, A252782, A292194.

Table[SeriesCoefficient[1/(1 - Sum[k^n x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - PolyLog[-n, x]), {x, 0, n}], {n, 0, 15}]

a(n) = [x^n] 1/(1 - PolyLog(-n,x)), where PolyLog() is the polylogarithm function.
From Vaclav Kotesovec, Mar 27 2018: (Start)
a(n) ~ 3^(n^2/3)                             if mod(n,3)=0
a(n) ~ 3^(n*(n-4)/3-2)*2^(2*n-1)*(n-1)*(n+8) if mod(n,3)=1
a(n) ~ 3^((n+1)*(n-3)/3)*2^n*(n+1)           if mod(n,3)=2
(End)

A320251 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 8, 8, 1, 1, 9, 18, 21, 16, 1, 1, 17, 44, 63, 55, 32, 1, 1, 33, 114, 207, 221, 144, 64, 1, 1, 65, 308, 723, 991, 776, 377, 128, 1, 1, 129, 858, 2631, 4805, 4752, 2725, 987, 256, 1, 1, 257, 2444, 9843, 24655, 31880, 22769, 9569, 2584, 512

Offset: 0

Ilya Gutkovskiy, Oct 08 2018

A(n,k) is the invert transform of k-th powers evaluated at n.

			G.f. of column k: A_k(x) = 1 + x + (2^k + 1)*x^2 + (2^(k + 1) + 3^k + 1)*x^3 + (3*2^k + 2^(2*k + 1) + 2*3^k + 1)*x^4 + ...
Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   2,   3,    5,    9,    17,     33,  ...
   4,   8,   18,   44,   114,    308,  ...
   8,  21,   63,  207,   723,   2631,  ...
  16,  55,  221,  991,  4805,  24655,  ...

N. J. A. Sloane, Transforms

Columns k=0..3 give A011782, A088305, A033453, A144109.
Main diagonal gives A301655.
Cf. A144048.

Table[Function[k, SeriesCoefficient[1/(1 - Sum[i^k x^i, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - PolyLog[-k, x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 - PolyLog(-k,x)), where PolyLog() is the polylogarithm function.

cp's OEIS Frontend

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

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A301655 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k).

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A320251 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).

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cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A301655 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A320251 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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