A007840 -id:A007840 - cp's OEIS frontend

A088815 Expansion of e.g.f. (1-x)^(-1/(1+log(1-x))).

1, 1, 4, 24, 190, 1860, 21638, 291158, 4443556, 75779580, 1427272032, 29409572808, 657829667328, 15868725580344, 410543007882408, 11336582934052104, 332736828827893968, 10342443317857993680, 339343476195341474688

Offset: 0

Vladeta Jovovic, Nov 22 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 0..410

Row sums of A079640.

Cf. A000262, A007840, A088819.

With[{nn=20},CoefficientList[Series[(1-x)^(-1/(1+Log[1-x])), {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Nov 29 2011 *)

x='x+O('x^25); Vec(serlaplace((1-x)^(-1/(1+log(1-x))))) \\ G. C. Greubel, Feb 16 2017

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, k!*abs(stirling(j, k, 1)))*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, May 23 2022

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A000262(k). - Vladeta Jovovic, Nov 26 2003
a(n) ~ n! * exp(n + 2*sqrt(n)/sqrt(exp(1)-1) + 1/(2*(exp(1)-1)) - 1/2) / (2*sqrt(Pi) * (exp(1)-1)^(n+1/4) * n^(3/4)). - Vaclav Kotesovec, May 04 2015
a(0) = 1; a(n) = Sum_{k=1..n} A007840(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, May 23 2022

A215916 The total number of components (cycles) in all alignments.

Original entry on oeis.org

0, 1, 5, 32, 254, 2414, 26746, 338568, 4820952, 76270032, 1327263024, 25196689968, 518190651744, 11476753967184, 272339818023984, 6893370154797312, 185387657162396544, 5279022594143270784, 158674547929990485888, 5020389181983702415104, 166784921186052433648896

Offset: 0

Geoffrey Critzer, Aug 27 2012

nonn

An alignment is a sequence of cycles of an n-permutation, cf. A007840.

Seiichi Manyama, Table of n, a(n) for n = 0..417
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 180.

Cf. A007840, A132393.

nn = 20; a = Log[1/(1 - x)];Range[0, nn]! CoefficientList[
  D[Series[1/(1 - y a), {x, 0, nn}], y] /. y -> 1, x]

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N, k*(-log(1-x))^k)))) \\ Seiichi Manyama, Apr 22 2022

a(n) = Sum_{k=1...n} s(n,k)*k!*k where s(n,k) is the unsigned Stirling number of the first kind (A132393).
E.g.f.: log(1/(1-x))/(1-log(1/(1-x)))^2.
a(n) ~ n!*n*exp(n)/(exp(1)-1)^(n+2) . - Vaclav Kotesovec, Sep 24 2013
E.g.f.: Sum_{k>=0} k * (-log(1-x))^k. - Seiichi Manyama, Apr 22 2022

A305323 Expansion of e.g.f. 1/(1 + log(1 + log(1 - x))).

Original entry on oeis.org

1, 1, 4, 25, 211, 2238, 28560, 425808, 7261200, 139367278, 2973006344, 69775267186, 1786673529746, 49565881948204, 1480900541242572, 47407364553205448, 1618838460981098680, 58734896900587841824, 2256402484187691207152, 91499934912942249975504, 3905739517580787866827872

Offset: 0

Ilya Gutkovskiy, May 30 2018

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 25*x^3/3! + 211*x^4/4! + 2238*x^5/5! + 28560*x^6/6! + ...

Robert Israel, Table of n, a(n) for n = 0..399

Cf. A003713, A007840, A217033, A305988.

S:= series(1/(1+log(1+log(1-x))),x,31):
seq(coeff(S,x,n)*n!,n=0..30); # Robert Israel, May 31 2018

nmax = 20; CoefficientList[Series[1/(1 + Log[1 + Log[1 - x]]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Sum[(j - 1)! Abs[StirlingS1[k, j]], {j, 1, k}] a[n - k]/k!, {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

x = 'x + O('x^30); Vec(serlaplace(1/(1 + log(1 + log(1 - x))))) \\ Michel Marcus, May 31 2018

a(n) ~ n! / (exp(2 - exp(-1)) * (1 - exp(exp(-1) - 1))^(n+1)). - Vaclav Kotesovec, May 31 2018

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007840(k). - Seiichi Manyama, May 11 2023

A305988 Expansion of e.g.f. 1/(1 + log(2 - exp(x))).

Original entry on oeis.org

1, 1, 4, 24, 194, 1970, 24062, 343294, 5601122, 102847794, 2098766582, 47117285270, 1154031484586, 30622256174458, 875092190716382, 26794239236959806, 875110094707912562, 30367988674208286914, 1115822099409002188358, 43276913813553367194598, 1766830322476935945014330

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A007840.

			1/(1 + log(2 - exp(x))) = 1 + x + 4*x^2/2! + 24*x^3/3! + 194*x^4/4! + 1970*x^5/5! + 24062*x^6/6! + ...

Robert Israel, Table of n, a(n) for n = 0..402
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000629, A000670, A007840, A050351, A083355, A305323.

b:= proc(n) b(n):= n!*`if`(n=0, 1, add(b(k)/(k!*(n-k)), k=0..n-1)) end:
a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nmax = 20; CoefficientList[Series[1/(1 + Log[2 - Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[StirlingS2[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k)*A007840(k).

a(n) ~ n! / ((2*exp(1) - 1) * (log(2 - exp(-1)))^(n+1)). - Vaclav Kotesovec, Jul 01 2018

A317171 a(n) = n! * [x^n] 1/(1 + n*log(1 - x)).

Original entry on oeis.org

1, 1, 10, 222, 8824, 553870, 50545008, 6328330344, 1041597412224, 218138133235680, 56650689388344000, 17868469522986145536, 6728682216722958185472, 2981868816113406609186576, 1536217706761623823662025728, 910442461680276910819097616000, 615053979239579281793375485526016

Offset: 0

Ilya Gutkovskiy, Jul 23 2018

nonn

Cf. A007840, A088500, A094420, A242817, A317172.

Main diagonal of A320079.

Table[n! SeriesCoefficient[1/(1 + n Log[1 - x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] n^k k!, {k, n}], {n, 16}]]

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

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n - 1/2). - Vaclav Kotesovec, Jul 23 2018

A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3n) Stirling1(n,k).

Original entry on oeis.org

1, 1, 129, 121172, 421875178, 3922823960054, 80130334773241142, 3156849112458066440568, 218554371053209725986724984, 24795129220015277612148345850896, 4365539219231132131300647267518575008, 1141930521329052244894253748456776246166288

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

In general, for m >= 0, Sum_{k=0..n} (-1)^(n-k) * k! * k^(m*n) * Stirling1(n,k) ~ c * r^(m*n) * (1 + r*exp(m/r))^n * n^((m+1)*n + 1/2) / exp((m+1)*n), where r is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-m/r) and c is a constant (depending only on m). - Vaclav Kotesovec, Feb 04 2022

Seiichi Manyama, Table of n, a(n) for n = 0..125

Cf. A007840 (m=0), A320096 (m=1), A351136 (m=2).

Cf. A203798, A351134, A351138.

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(3*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 12, 0] (* Amiram Eldar, Feb 02 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^3*x))^k)))

E.g.f.: Sum_{k>=0} (-log(1 - k^3*x))^k.

a(n) ~ c * r^(3*n) * (1 + r*exp(3/r))^n * n^(4*n + 1/2) / exp(4*n), where r = 0.97698437755148201976772582981871258235824532360125531194... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-3/r) and c = 2.3655154360078103511101518906595610482889989819... - Vaclav Kotesovec, Feb 04 2022

A351138 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(kn) Stirling1(n,k).

Original entry on oeis.org

1, 1, 33, 118484, 103098352618, 35763050751038414134, 7426387531294394110580641088438, 1294894837982331434068068403253026516109577144, 253092742000650212462862632240661689524832716838851180353875064

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A007840, A320096, A351136, A351137.

Cf. A249584, A351135.

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(k*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 9, 0] (* Amiram Eldar, Feb 02 2022 *)

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

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^k*x))^k)))

E.g.f.: Sum_{k>=0} (-log(1 - k^k*x))^k.

a(n) ~ n! * n^(n^2). - Vaclav Kotesovec, Feb 03 2022

A352071 Expansion of e.g.f. 1 / (1 + log(1 - 4*x) / 4).

Original entry on oeis.org

1, 1, 6, 62, 904, 16984, 390128, 10586736, 331267200, 11738697600, 464539452672, 20302660659456, 971106358760448, 50452643588275200, 2829000818124208128, 170271405502300207104, 10948525752699316371456, 748994717201835804033024, 54315931193865932254543872

Offset: 0

Ilya Gutkovskiy, Mar 02 2022

nonn

Cf. A007840, A047053, A210246, A227917, A326324, A352069.

nmax = 18; CoefficientList[Series[1/(1 + Log[1 - 4 x]/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-4)^(n - k), {k, 0, n}], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-4*x)/4))) \\ Michel Marcus, Mar 02 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-4)^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * 4^(k-1) * a(n-k).
a(n) ~ n! * 4^(n+1) * exp(4*n) / (exp(4) - 1)^(n+1). - Vaclav Kotesovec, Mar 03 2022

A355087 E.g.f. A(x) satisfies A(x) = 1 - log(1-x) * A(3*x).

Original entry on oeis.org

1, 1, 7, 200, 22008, 8968614, 13103201154, 66911528280456, 1170951806182816008, 69148802888161984160496, 13610904625275511792310726256, 8840888240006182011615570947541552, 18793694994107665289261434484161549856688

Offset: 0

Seiichi Manyama, Jun 18 2022

nonn

Cf. A007840, A355086.

Cf. A355088.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, 3^(i-j)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

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

A355284 Expansion of e.g.f. 1 / (1 + x + x^2/2 + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 2, 6, 24, 200, 1560, 12936, 130368, 1458432, 17623440, 233922480, 3376625472, 52382131776, 870882440064, 15459372915840, 291596692838400, 5824039155720192, 122814724467223296, 2726547887891407104, 63562453551393223680, 1552499303360183700480

Offset: 0

Ilya Gutkovskiy, Jun 26 2022

nonn

Cf. A007840, A038205, A102233, A226226, A355285.

nmax = 22; CoefficientList[Series[1/(1 + x + x^2/2 + 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, 3, n}]; Table[a[n], {n, 0, 22}]

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

E.g.f.: 1 / (1 - Sum_{k>=3} x^k/k).

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

cp's OEIS Frontend

A088815 Expansion of e.g.f. (1-x)^(-1/(1+log(1-x))).

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A215916 The total number of components (cycles) in all alignments.

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A305323 Expansion of e.g.f. 1/(1 + log(1 + log(1 - x))).

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A305988 Expansion of e.g.f. 1/(1 + log(2 - exp(x))).

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A317171 a(n) = n! * [x^n] 1/(1 + n*log(1 - x)).

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A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3*n) * Stirling1(n,k).

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A351138 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(k*n) * Stirling1(n,k).

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A352071 Expansion of e.g.f. 1 / (1 + log(1 - 4*x) / 4).

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A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3n) Stirling1(n,k).

A351138 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(kn) Stirling1(n,k).