A052811 -id:A052811 - cp's OEIS frontend

A005359 a(n) = n! if n is even, otherwise 0 (from Taylor series for cos x).

1, 0, 2, 0, 24, 0, 720, 0, 40320, 0, 3628800, 0, 479001600, 0, 87178291200, 0, 20922789888000, 0, 6402373705728000, 0, 2432902008176640000, 0, 1124000727777607680000, 0, 620448401733239439360000, 0

Offset: 0

N. J. A. Sloane, Russ Cox

nonn

Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.
Stirling transform of a(n)=[0,2,0,24,0,720,...] is A052841(n)=[0,2,6,38,270,...]. - Michael Somos, Mar 04 2004
Stirling transform of a(n-1)=[1,0,2,0,24,0,...] is A000670(n-1)=[1,1,3,13,75,...]. - Michael Somos, Mar 04 2004
Stirling transform of a(n-1)=[0,0,2,0,24,0,...] is A052875(n-1)=[0,0,2,12,74,...]. - Michael Somos, Mar 04 2004
Stirling transform of (-1)^n*A052811(n)=[0,2,-6,46,-340,...] is a(n)=[0,2,0,24,0,...]. - Michael Somos, Mar 04 2004
Also n-th derivative of arctanh(x) at x=0. - Michel Lagneau, Aug 13 2012
Binomial convolution square of A177145 (with offset 0) because each permutation in S_{2n} uniquely determines a bi-partition of its elements into even and odd cycles and these are both enumerated by A177145. - Michael Somos, Mar 19 2019

Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael Somos, Number of permutations with all cycles of even length, answer to Mathematics Stack Exchange question 3152701, Mar 18 2019.
Index entries for sequences related to factorial numbers

From Johannes W. Meijer, Nov 12 2009: (Start)
Equals the first right hand column of A167565.
Equals the first left hand column of A167568.
(End)
Cf. A177145.
Bisection (even part) gives A010050.

BB:={E=Prod(Z,Z),S=Union(Epsilon,Prod(S,E))}: ZL:=[S,BB, labeled]: > seq(count(ZL,size=n),n=0..25); # Zerinvary Lajos, Apr 22 2007
a:=n->n!+(-1)^n*n!: seq(a(n)/2, n=0..25); # Zerinvary Lajos, Mar 25 2008

Riffle[Range[0,30,2]!,0] (* Harvey P. Dale, Nov 16 2011 *)
a[ n_] := If[n >= 0 && EvenQ[n], n!, 0]; (* Michael Somos, Mar 19 2019 *)

{a(n) = if(n<0, 0, if(n%2, 0, n!))}; /* Michael Somos, Mar 04 2004 */

E.g.f. 1/(1-x^2) = d/dx log(sqrt((1+x)/(1-x))). a(2n)=(2n)!, a(2n+1)=0. - Michael Somos, Mar 04 2004
a(n) = Product_{k=0..n/2-1} binomial(n-2k,2)*2^(n/2) for even n. - Geoffrey Critzer, Jun 05 2016
From Ilya Gutkovskiy, Jun 05 2016: (Start)
D-finite with recurrence a(n) = n*(n - 1)*a(n-2), a(0)=1, a(1)=0.
a(n) = n!*((-1)^n + 1)/2. (End)

A052801 A simple grammar: labeled pairs of sequences of cycles.

Original entry on oeis.org

1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856, 89379120, 1534623936, 28804923024, 586686138384, 12885385945248, 303537419684064, 7633673997722496, 204125888803996800, 5782960189212871680

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Robert Israel, Table of n, a(n) for n = 0..417
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 759.

Cf. A215916.

Cf. A007840, A354122, A354123.

spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/(1+Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

makelist(sum((-1)^(n-k)*stirling1(n, k)*(k+1)!, k, 0, n), n, 0, 17); /* Bruno Berselli, May 25 2011 */

E.g.f.: 1/(-1+log(-1/(-1+x)))^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*(k+1)!. - Vladeta Jovovic, Sep 21 2003
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A052811. - Peter Bala, Nov 25 2011
a(n) ~ n! * n*exp(n)/(exp(1)-1)^(n+2). - Vaclav Kotesovec, Sep 30 2013
From Anton Zakharov, Aug 07 2016: (Start)
a(n) = A007840(n) + A215916(n).
a(n) = Sum_{k=2..n+1} k!*s(n,k) where s(n,k) is the unsigned Stirling number of the first kind, (A132393). (End)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A346921 Expansion of e.g.f. 1 / (1 - log(1 - x)^2 / 2).

Original entry on oeis.org

1, 0, 1, 3, 17, 110, 874, 8064, 85182, 1012248, 13369026, 194245590, 3079135806, 52880064588, 978038495316, 19381794788160, 409702099828104, 9201877089355584, 218832476773294008, 5493266481129425064, 145153549897858762776, 4027310838211114515600

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..430

Cf. A007840, A346922, A346923, A346924.

Cf. A000254, A052811, A305306, A330047, A347001.

nmax = 21; CoefficientList[Series[1/(1 - Log[1 - x]^2/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^2/2))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/2^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,2)| * a(n-k).
a(n) ~ n! * exp(sqrt(2)*n) / (sqrt(2) * (exp(sqrt(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/2^k. - Seiichi Manyama, May 06 2022

A353118 Expansion of e.g.f. 1/(1 + log(1 - x)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 2070, 24864, 310632, 4337544, 68922360, 1205002656, 22844264256, 469287123552, 10397824478496, 246800350393344, 6246190572981120, 167972669001740160, 4783274802508890240, 143775432034543203840, 4548946867429143444480

Offset: 0

Seiichi Manyama, May 06 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..418

Cf. A052811, A353119, A353200.

Cf. A346922.

With[{nn=20},CoefficientList[Series[1/(1+Log[1-x]^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 04 2023 *)

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*abs(stirling(j, 3, 1))*v[i-j+1])); v;

a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1)));

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A353200 Expansion of e.g.f. 1/(1 + log(1 - x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 35947800, 609615600, 12504927600, 281242996320, 6545492073120, 155873050569600, 3849612346944000, 100588974863402880, 2818516832681523840, 84728757269204858880, 2706516690047188416000

Offset: 0

Seiichi Manyama, May 06 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..418

Cf. A052811, A353118, A353119.

Cf. A346924.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^5)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*abs(stirling(j, 5, 1))*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1)));

a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A353119 Expansion of e.g.f. 1/(1 - log(1 - x)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 2040, 17640, 202776, 3066336, 52446720, 933636000, 17416490784, 350580364992, 7719355635264, 184232862777600, 4691944607751936, 126358891201529856, 3591751011211717632, 107772466927523060736, 3408777017097439186944

Offset: 0

Seiichi Manyama, May 06 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..418

Cf. A052811, A353118, A353200.

Cf. A346923.

With[{nn=20},CoefficientList[Series[1/(1-Log[1-x]^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 13 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1-x)^4)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i, j)*abs(stirling(j, 4, 1))*v[i-j+1])); v;

a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1)));

a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! |Stirling1(n,k*j)|.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 14, 0, 1, 0, 0, 6, 88, 0, 1, 0, 0, 6, 46, 694, 0, 1, 0, 0, 0, 36, 340, 6578, 0, 1, 0, 0, 0, 24, 210, 3308, 72792, 0, 1, 0, 0, 0, 0, 240, 2070, 36288, 920904, 0, 1, 0, 0, 0, 0, 120, 2040, 24864, 460752, 13109088, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 310632, 6551424, 207360912, 0

Offset: 0

Seiichi Manyama, Oct 18 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   0,   0,   0,   0, ...
  0,   3,   2,   0,   0,   0, ...
  0,  14,   6,   6,   0,   0, ...
  0,  88,  46,  36,  24,   0, ...
  0, 694, 340, 210, 240, 120, ...

Columns k=0-5 give: A000007, A007840, A052811, A353118, A353119, A353200.

Cf. A357119, A357868, A357882.

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

T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(-log(1-x+x*O(x^n)))^k), n));

For k > 0, e.g.f. of column k: 1/(1 - (-log(1-x))^k).

T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * |Stirling1(j,k)| * T(n-j,k).

cp's OEIS Frontend

A005359 a(n) = n! if n is even, otherwise 0 (from Taylor series for cos x).

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A052801 A simple grammar: labeled pairs of sequences of cycles.

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A346921 Expansion of e.g.f. 1 / (1 - log(1 - x)^2 / 2).

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A353118 Expansion of e.g.f. 1/(1 + log(1 - x)^3).

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A353200 Expansion of e.g.f. 1/(1 + log(1 - x)^5).

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A353119 Expansion of e.g.f. 1/(1 - log(1 - x)^4).

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A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|.

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A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! |Stirling1(n,k*j)|.