A007113 -id:A007113 - cp's OEIS frontend

A354612 Expansion of e.g.f. exp(f(x) - 1) where f(x) = (1 + x)^x = e.g.f. for A007113.

1, 0, 2, -3, 32, -150, 1404, -11340, 120448, -1319976, 16600320, -223664760, 3300331704, -52223268240, 887583503520, -16071609481200, 309263446333440, -6296705309543040, 135262191966465600, -3056359409652695040, 72462969268541596800

Offset: 0

Seiichi Manyama, Jul 08 2022

sign

Cf. A007113, A202152, A354613.

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

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

a(0) = 1; a(n) = Sum_{k=1..n} A007113(k) * binomial(n-1,k-1) * a(n-k).

A066166 Stanley's children's game. Class of n (named) children forms into rings with exactly one child inside each ring. We allow the case when outer ring has only one child. a(n) gives number of possibilities, including clockwise order (or which hand is held), in each ring.

Original entry on oeis.org

2, 3, 20, 90, 594, 4200, 34544, 316008, 3207240, 35699400, 432690312, 5672581200, 79991160144, 1207367605080, 19423062612480, 331770360922560, 5997105160795584, 114373526841360000, 2295170834453089920

Offset: 2

Len Smiley, Dec 12 2001

Apparently n divides a(n), so a(n)/n = 1, 1, 5, 18, 99, 600, 4318, 35112, 320724, 3245400, 36057526, 436352400, 5713654296, ... - R. J. Mathar, Oct 31 2015

			a(4)=20: 12 ways to make 2 hugs, 8 ways to make a 3-ring.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Harvey P. Dale, Table of n, a(n) for n = 2..250
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function, arXiv:math/0501379 [math.CO], 2005.

Cf. A066165. Apart from initial terms and signs, same as A007113.

Cf. A343579.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-1+1/(1-x)^x)); [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Aug 29 2018

Drop[With[{nn=20},CoefficientList[Series[1/(1-x)^x-1,{x,0,nn}],x] Range[ 0,nn]!],2] (* Harvey P. Dale, Sep 17 2011 *)

b(n):=if n=0 then 1 else (n-1)!*sum((1+1/i)*b(n-i-1)/(n-i-1)!,i,1,n-1);
makelist(a(n),n,2,10); /* Vladimir Kruchinin, Feb 25 2015 */

a(n)=if(n<0,0,n!*polcoeff(-1+1/(1-x+x*O(x^n))^x,n))

{a(n) = n!*polcoeff( sum(m=1,n, x^m/m! * prod(k=0,m-1,x + k) +x*O(x^n) ), n)}
for(n=2,20, print1(a(n),", ")) \\ Paul D. Hanna, Oct 26 2015

a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 10 2022

E.g.f.: -1+1/(1-x)^x.
a(n) ~ n! * (1 - 1/n + (1-log(n)-gamma)/n^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014
a(n) = b(n), n>0, a(0)=0, where b(n) = (n-1)!*Sum_{i=1..n-1} (1+1/i)*b(n-i-1)/(n-i-1)!, b(0)=1. - Vladimir Kruchinin, Feb 25 2015
E.g.f.: Sum_{n>=1} x^n/n! * Product_{k=0..n-1} (k + x). - Paul D. Hanna, Oct 26 2015
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, May 10 2022

A053489 Expansion of e.g.f.: (1-x)^(-2x).

Original entry on oeis.org

1, 0, 4, 6, 64, 300, 2568, 20160, 193856, 1989792, 22687200, 279956160, 3737966208, 53589444480, 821522026752, 13407498599040, 232106716968960, 4248256958023680, 81968803604600832, 1662870215019018240, 35384007384670648320, 788053048823608565760

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.3.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A007113, A053490.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^(-2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 29 2018

CoefficientList[Series[(1-x)^(-2*x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 21 2014 *)

x='x+O('x^30); Vec(serlaplace((1-x)^(-2*x))) \\ G. C. Greubel, Aug 29 2018

E.g.f.: (1-x)^(-2*x).
a(n) = (-1)^n*Sum_{k=0..floor(n/2)} 2^k*binomial(n, k)*k!*Stirling1(n-k, k). - Vladeta Jovovic, Dec 19 2004
a(n) ~ n! * n * (1 + (1-2*log(n)-2*gamma)/n), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014

A053490 Expansion of e.g.f.: (1-x)^(-3x).

Original entry on oeis.org

1, 0, 6, 9, 132, 630, 6642, 55440, 608976, 6790392, 85413960, 1145077560, 16600386888, 256806229680, 4233767671728, 74015194485960, 1368023697469440, 26649263762049600, 545697922821501888, 11717708270380421760, 263276186128105633920

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.3.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A007113, A053489.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)^(-3*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 29 2018

CoefficientList[Series[(1-x)^(-3*x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 21 2014 *)

x='x+O('x^30); Vec(serlaplace((1-x)^(-3*x))) \\ G. C. Greubel, Aug 29 2018

a(n) = (-1)^n*Sum_{k=0..floor(n/2)} 3^k*binomial(n, k)*k!*Stirling1(n-k, k). - Vladeta Jovovic, Dec 19 2004

a(n) ~ n! * n^2/2 * (1 + (9-6*log(n)-6*gamma)/n), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014

A349559 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^x.

Original entry on oeis.org

1, 0, 2, 3, 44, 270, 3714, 44940, 746528, 13149864, 271954440, 6154715160, 155055594792, 4254730262640, 127019898548256, 4088313657038520, 141301521555548160, 5216698433745207360, 204946906542573645504, 8536144551987171202560

Offset: 0

Seiichi Manyama, Nov 22 2021

nonn

Cf. A007113, A349556.

Cf. A184949, A356786, A356787.

a:= n-> n!*coeff(series(RootOf(1/(1-x*A)^x-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 22 2021

nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x*A[x])^x + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)

my(A=1,n=22); for(i=1, n, A=(1-x*A)^(-x+x*O(x^n))); Vec(serlaplace(A))

a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, Aug 27 2022

a(n) ~ sqrt(1 + r - 2*r*log(r)) * n^(n-1) / ((1+r)^2 * exp(n) * r^(n + 1/2)), where r = 0.4214518303433019663622598075106479936652984008256... is the root of the equation r^(1-r) * (1+r)^(1+r) = 1. - Vaclav Kotesovec, Nov 22 2021

a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Aug 27 2022

A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 6, 20, 0, 1, 0, 0, 0, -12, -90, 0, 1, 0, 0, 0, 24, 40, 594, 0, 1, 0, 0, 0, 0, -60, 180, -4200, 0, 1, 0, 0, 0, 0, 120, 240, -1512, 34544, 0, 1, 0, 0, 0, 0, 0, -360, -1260, 11760, -316008, 0, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, -38880, 3207240, 0

Offset: 0

Seiichi Manyama, Jul 09 2022

			Square array begins:
  1,   1,   1,   1,    1,   1, 1, ...
  1,   0,   0,   0,    0,   0, 0, ...
  0,   2,   0,   0,    0,   0, 0, ...
  0,  -3,   6,   0,    0,   0, 0, ...
  0,  20, -12,  24,    0,   0, 0, ...
  0, -90,  40, -60,  120,   0, 0, ...
  0, 594, 180, 240, -360, 720, 0, ...

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

Columns k=1..4 give A007113, A007121, (-1)^n * A353229(n), A354625.

Cf. A292892, A355609, A355619.

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

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

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

A053491 Expansion of e.g.f. (1-2*x)^(-x).

Original entry on oeis.org

1, 0, 4, 12, 112, 960, 10848, 141120, 2122496, 36094464, 685578240, 14385761280, 330532435968, 8253827112960, 222587077558272, 6447285982126080, 199630453605335040, 6580280144225894400, 230056747973625249792, 8503148524089755566080

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A007113, A053489, A066166.

CoefficientList[Series[(1-2x)^(-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

a(n) = n!*sum(k=0, n\2, 2^(n-k)*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 20 2022

a(n) ~ 2^(n+1/2)*n^n/exp(n). - Vaclav Kotesovec, Jun 27 2013

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, May 20 2022

A318615 a(n) = n! * [x^n] 1/(1 - x)^(n*x).

Original entry on oeis.org

1, 0, 4, 9, 224, 1650, 38664, 540960, 13930496, 291769128, 8598924000, 237964577400, 8082061452288, 275311724996880, 10714824398213376, 430458433091505000, 19007133744632954880, 876046954673290438080, 43416883192646088235008, 2252711496770428822876800, 124040138653975179571200000

Offset: 0

Ilya Gutkovskiy, Aug 30 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A007113, A008275, A053489, A053490, A191415, A318616.

Table[n! SeriesCoefficient[1/(1 - x)^(n x), {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[(-1)^n n! Sum[n^(n - k) StirlingS1[k, n - k]/k!, {k, n}], {n, 20}]]

a(n) = n! * [x^n] exp(n*x*Sum_{k>=1} x^k/k).
a(n) = (-1)^n*n! * Sum_{k=0..n} n^(n-k)*Stirling1(k,n-k)/k!.
a(n) ~ n^n / (sqrt(1 - (1-s)*(2-s)*s) * exp(n) * s^n * (1-s)^(s*n - 1)), where s = 0.530402312512063468084914246777198746... is the root of the equation (1-s)*(2 + s + s*log(1-s)) = 1. - Vaclav Kotesovec, Aug 30 2018

A355603 Expansion of e.g.f. (1 + x)^(x^4/24).

Original entry on oeis.org

1, 0, 0, 0, 0, 5, -15, 70, -420, 3024, -22050, 202950, -2113650, 24324300, -305645340, 4174483950, -61253992800, 961049212200, -16054949350440, 284505099278400, -5329752594075000, 105239780964864000, -2184466455408699000, 47550052231211237400

Offset: 0

Seiichi Manyama, Jul 09 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A007113, A355605.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^(x^4/24)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/24*log(1+x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!/24*sum(j=5, i, (-1)^j*j/(j-4)*v[i-j+1]/(i-j)!)); v;

a(n) = n!*sum(k=0, n\5, stirling(n-4*k, k, 1)/(24^k*(n-4*k)!));

a(0) = 1; a(n) = -(n-1)!/24 * Sum_{k=5..n} (-1)^k * k/(k-4) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/5)} Stirling1(n-4*k,k)/(24^k * (n-4*k)!).

A355619 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k/k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 3, 20, 0, 1, 0, 0, 0, -6, -90, 0, 1, 0, 0, 0, 4, 20, 594, 0, 1, 0, 0, 0, 0, -10, 0, -4200, 0, 1, 0, 0, 0, 0, 5, 40, -126, 34544, 0, 1, 0, 0, 0, 0, 0, -15, -210, 1260, -316008, 0, 1, 0, 0, 0, 0, 0, 6, 70, 1904, -4320, 3207240, 0

Offset: 0

Seiichi Manyama, Jul 10 2022

			Square array begins:
  1,   1,  1,   1,   1, 1, 1, ...
  1,   0,  0,   0,   0, 0, 0, ...
  0,   2,  0,   0,   0, 0, 0, ...
  0,  -3,  3,   0,   0, 0, 0, ...
  0,  20, -6,   4,   0, 0, 0, ...
  0, -90, 20, -10,   5, 0, 0, ...
  0, 594,  0,  40, -15, 6, 0, ...

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

Columns k=1..4 give A007113, A355605, (-1)^n * A351493(n), A355603.

Cf. A355607, A355610.

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

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

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

cp's OEIS Frontend

A354612 Expansion of e.g.f. exp(f(x) - 1) where f(x) = (1 + x)^x = e.g.f. for A007113.

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A066166 Stanley's children's game. Class of n (named) children forms into rings with exactly one child inside each ring. We allow the case when outer ring has only one child. a(n) gives number of possibilities, including clockwise order (or which hand is held), in each ring.

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A053489 Expansion of e.g.f.: (1-x)^(-2x).

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A053490 Expansion of e.g.f.: (1-x)^(-3x).

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A349559 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^x.

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A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

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A053491 Expansion of e.g.f. (1-2*x)^(-x).

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