A007840 -id:A007840 - cp's OEIS frontend

A337060 E.g.f.: 1 / (1 + x^2/2 + log(1 - x)).

1, 1, 2, 8, 46, 324, 2708, 26424, 295272, 3714600, 51929472, 798610416, 13399081584, 243556758912, 4767863027328, 100004300847744, 2237419620187776, 53187370914349440, 1338737435337261312, 35568441673932566016, 994744655047298951424, 29211127285363209561600

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

nonn

Cf. A000266, A007840, A226226, A337061.

nmax = 21; CoefficientList[Series[1/(1 + x^2/2 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 21}]

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=3..n} binomial(n,k) * (k-1)! * a(n-k).

A337061 E.g.f.: 1 / (1 + x^3/3 + log(1 - x)).

Original entry on oeis.org

1, 1, 3, 12, 72, 534, 4818, 50532, 606408, 8182656, 122712912, 2024328096, 36432644400, 710346495312, 14915647605168, 335567743462944, 8052843408926976, 205328108580310656, 5543345188496499840, 157970863597032124416, 4738694884696030305024

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

nonn

Cf. A000090, A007840, A226226, A337060.

nmax = 20; CoefficientList[Series[1/(1 + x^3/3 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[n_] := a[n] = n (a[n - 1] + (n - 1) a[n - 2]/2) + Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 20}]

a(0) = a(1) = 1; a(n) = n * (a(n-1) + (n-1) * a(n-2) / 2) + Sum_{k=4..n} binomial(n,k) * (k-1)! * a(n-k).

A346432 a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.

Original entry on oeis.org

1, 2, 14, 144, 1968, 33600, 688320, 16450560, 449326080, 13806858240, 471395635200, 17703899136000, 725338710835200, 32193996432998400, 1538840509503897600, 78808952068374528000, 4305129487814098944000, 249876735246162984960000, 15356385691181506363392000

Offset: 0

Ilya Gutkovskiy, Jul 17 2021

nonn

Cf. A000670, A001339, A002866, A003480, A007840, A052555, A052567, A136658, A216794, A308939, A346433.

a[0] = 1; a[n_] := a[n] = n! Sum[(n - k + 1) a[k]/k!, {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 1/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) StirlingS1[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - 1 / (1 - x)^2))) \\ Michel Marcus, Jul 18 2021

E.g.f.: 1 / (2 - 1 / (1 - x)^2).
E.g.f.: 1 / (1 - Sum_{k>=1} (k+1) * x^k).
a(0) = 1, a(1) = 2, a(2) = 14; a(n) = 4 * n * a(n-1) - 2 * n * (n-1) * a(n-2).
a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * 2^k * A000670(k).
a(n) = n! * A003480(n).

A347949 E.g.f.: 1 / (1 - Sum_{k>=1} x^prime(k) / prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 6, 64, 170, 2988, 14616, 180192, 1934712, 21673200, 300266736, 4220710272, 61785461712, 1003589762784, 17448621367680, 327598207658496, 6279739240655232, 134169095009652480, 2817563310900129024, 64570676279407718400, 1547773850801172960000, 38824156236466815920640

Offset: 0

Ilya Gutkovskiy, Sep 20 2021

nonn

Cf. A007840, A010051, A023360, A218002, A347948.

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * A010051(k) * a(n-k).

A355285 Expansion of e.g.f. 1 / (1 + x + x^2/2 + x^3/3 + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 6, 24, 120, 720, 7560, 76608, 810432, 9141120, 118015920, 1666336320, 25211774016, 404932155264, 6951992261760, 127203705538560, 2467434718218240, 50477473338494976, 1086707769452699904, 24573149993692615680, 582367494447600583680, 14430857455114783119360

Offset: 0

Ilya Gutkovskiy, Jun 26 2022

nonn

Cf. A007840, A047865, A226226, A232475, A355284.

nmax = 23; CoefficientList[Series[1/(1 + x + x^2/2 + x^3/3 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(1/(1 + x + x^2/2 + x^3/3 + log(1 - x)))) \\ Michel Marcus, Jun 27 2022

E.g.f.: 1 / (1 - Sum_{k>=4} x^k/k).

a(0) = 1; a(n) = Sum_{k=4..n} binomial(n,k) * (k-1)! * a(n-k).

A355665 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k * log(1 - x)).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 14, 1, 0, 0, 3, 88, 1, 0, 0, 6, 32, 694, 1, 0, 0, 0, 12, 150, 6578, 1, 0, 0, 0, 24, 40, 1524, 72792, 1, 0, 0, 0, 0, 60, 900, 12600, 920904, 1, 0, 0, 0, 0, 120, 240, 6048, 147328, 13109088, 1, 0, 0, 0, 0, 0, 360, 1260, 43680, 1705536, 207360912

Offset: 0

Seiichi Manyama, Jul 13 2022

			Square array begins:
     1,    1,   1,   1,   1,   1, 1, ...
     1,    0,   0,   0,   0,   0, 0, ...
     3,    2,   0,   0,   0,   0, 0, ...
    14,    3,   6,   0,   0,   0, 0, ...
    88,   32,  12,  24,   0,   0, 0, ...
   694,  150,  40,  60, 120,   0, 0, ...
  6578, 1524, 900, 240, 360, 720, 0, ...

Columns k=0..3 give A007840, A052830, A351503, A351504.

Cf. A355609, A355652.

T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(n-k*j)!);

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

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(n-k*j)!.

A368284 Expansion of e.g.f. exp(-2*x) / (1 + log(1 - x)).

Original entry on oeis.org

1, -1, 3, 0, 32, 182, 1882, 20500, 260136, 3701968, 58565360, 1019110848, 19346296752, 397867297136, 8811800026928, 209100451072672, 5292665533921024, 142338738348972672, 4053176346277660288, 121828547313861426176, 3854597854165079424768

Offset: 0

Seiichi Manyama, Dec 19 2023

sign

Cf. A007840, A291979, A330149, A368283.

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-2)^i+sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

a(n) = (-2)^n + Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).

A368754 a(n) = (n!)^n * [x^n] * 1/(1 - polylog(n,x)).

Original entry on oeis.org

1, 1, 5, 278, 404768, 28436662624, 151309093659896512, 86745908552613198656020224, 7184659625769578063908866060107907072, 110866279942987479997999976181870531647691458347008, 399488258540989429698770032526869852804662313023226648081962369024

Offset: 0

Alois P. Heinz, Jan 04 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A000051, A000142, A007840, A011782, A036740, A323339, A323340, A336258, A336259, A336260, A336261.

a:= n-> n!^n*coeff(series(1/(1-polylog(n, x)), x, n+1), x, n):
seq(a(n), n=0..10);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)/j^k, j=1..n))
    end:
a:= n-> n!^n*b(n$2):
seq(a(n), n=0..10);

a(n) = (n!)^n*b(n,n) with b(n,k) = Sum_{j=1..n} b(n-j,k)/j^k for n>0, b(0,k) = 1.

A382808 a(n) = Sum_{k=0..n} (|Stirling1(n,k)| * k!)^3.

Original entry on oeis.org

1, 1, 9, 440, 71344, 25826824, 17321581592, 19304140340736, 33142988156751360, 82906630912116006912, 289508760665893747703808, 1364207202603804952193826816, 8438589244471363680258331914240, 66972265137135031645961782287814656, 668922701586813036491303458870218731520

Offset: 0

Ilya Gutkovskiy, Apr 05 2025

nonn

In general, for m>=1, Sum_{k=0..n} (abs(Stirling1(n,k)) * k!)^m ~ sqrt(2*Pi/m) * n^(m*n + 1/2) / (exp(1) - 1)^(m*n+1). - Vaclav Kotesovec, Apr 05 2025

Cf. A007840, A242280, A382792, A382807.

Table[Sum[(Abs[StirlingS1[n, k]] k!)^3, {k, 0, n}], {n, 0, 14}]
Table[(n!)^3 SeriesCoefficient[1/(1 + Log[1 - x] Log[1 - y] Log[1 - z]), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 14}]

a(n) = (n!)^3 * [(x*y*z)^n] 1 / (1 + log(1 - x) * log(1 - y) * log(1 - z)).

a(n) ~ sqrt(2*Pi/3) * n^(3*n + 1/2) / (exp(1) - 1)^(3*n+1). - Vaclav Kotesovec, Apr 05 2025

A382830 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * |Stirling1(n,k)| * k!.

Original entry on oeis.org

1, 1, 8, 102, 1804, 40890, 1131108, 36948240, 1391945616, 59411849040, 2833582748160, 149347596487056, 8620256620495584, 540775669746661440, 36636074309252234880, 2665704585421541790720, 207329122282259073044736, 17165075378189396045777280, 1507206260097615729874083840

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

nonn

Cf. A007840, A052801, A277759, A305919, A354122, A354123.

Table[Sum[Binomial[n + k - 1, k] Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[1/(1 + Log[1 - x])^n, {x, 0, n}], {n, 0, 18}]

a(n) = n! * [x^n] 1 / (1 + log(1 - x))^n.

a(n) ~ LambertW(exp(2))^n * n^n / (sqrt(1 + LambertW(exp(2))) * exp(n) * (LambertW(exp(2)) - 1)^(2*n)). - Vaclav Kotesovec, Apr 06 2025

cp's OEIS Frontend

A337060 E.g.f.: 1 / (1 + x^2/2 + log(1 - x)).

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A337061 E.g.f.: 1 / (1 + x^3/3 + log(1 - x)).

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A346432 a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.

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A347949 E.g.f.: 1 / (1 - Sum_{k>=1} x^prime(k) / prime(k)).

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A355285 Expansion of e.g.f. 1 / (1 + x + x^2/2 + x^3/3 + log(1 - x)).

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A355665 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k * log(1 - x)).

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PARI

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A368284 Expansion of e.g.f. exp(-2*x) / (1 + log(1 - x)).

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PARI

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A368754 a(n) = (n!)^n * [x^n] * 1/(1 - polylog(n,x)).

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Maple

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A382808 a(n) = Sum_{k=0..n} (|Stirling1(n,k)| * k!)^3.

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A382830 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * |Stirling1(n,k)| * k!.

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