A052802 -id:A052802 - cp's OEIS frontend

A370993 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + xlog(1-x)) ).

1, 0, 2, 3, 80, 450, 11424, 133140, 3670400, 68303088, 2123212320, 54742984560, 1938915574848, 63653459126400, 2565847637273088, 101718189575664480, 4637150408792355840, 214393171673968519680, 10962579011721928980480, 577166004742408670937600

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A052802, A370994, A370995.

Cf. A052830, A370988.

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

a(n) = sum(k=0, n\2, (n+k)!*abs(stirling(n-k, k, 1))/(n-k)!)/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (n+k)! * |Stirling1(n-k,k)|/(n-k)!.

A367139 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^3)).

Original entry on oeis.org

1, 1, 9, 167, 4780, 186004, 9173780, 548563140, 38573633016, 3119384230176, 285237426927552, 29102185296785160, 3277703460197645232, 403931173342682581296, 54066960915411480743520, 7811249803193620134996864, 1211525560869437165319590400

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A007840, A052802, A367138.

Cf. A367135, A367137.

Table[1/(3*n+1)! * Sum[(3*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

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

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * |Stirling1(n,k)|.

a(n) ~ LambertW(3*exp(4))^n * n^(n-1) / (sqrt(3*(1 + LambertW(3*exp(4)))) * exp(n) * (-3 + LambertW(3*exp(4)))^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367138 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^2)).

Original entry on oeis.org

1, 1, 7, 98, 2096, 60684, 2221766, 98488592, 5129567208, 307066395000, 20775900638472, 1567955813868960, 130596146677118448, 11899839375083061024, 1177540373453616858240, 125754589311488009416704, 14416305655742615673941760, 1765794816084642802179333120

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A007840, A052802, A367139.

Cf. A367134, A367136.

Table[1/(2*n+1)! * Sum[(2*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

a(n) = sum(k=0, n, (2*n+k)!*abs(stirling(n, k, 1)))/(2*n+1)!;

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * |Stirling1(n,k)|.

a(n) ~ LambertW(2*exp(3))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(3)))) * exp(n) * (-2 + LambertW(2*exp(3)))^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367159 E.g.f. satisfies A(x) = 1 - A(x)^2 * log(1 - x*A(x)).

Original entry on oeis.org

1, 1, 7, 95, 1954, 54244, 1901560, 80648658, 4016874920, 229881369768, 14866341101064, 1072223706468672, 85337672738960736, 7429736462231570304, 702426961910810154624, 71667022709644235679120, 7848761844632669045606016, 918383128248130459272478080

Offset: 0

Seiichi Manyama, Nov 07 2023

nonn

Cf. A052802, A138013, A367160.

a(n) = sum(k=0, n, (n+2*k)!/(n+k+1)!*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} (n+2*k)!/(n+k+1)! * |Stirling1(n,k)|.

A198860 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x))).

Original entry on oeis.org

1, 1, 3, 17, 144, 1634, 23312, 401274, 8096680, 187472136, 4900535832, 142766286552, 4587190461840, 161161214978880, 6146415080939520, 252902928346825104, 11167368115492742400, 526752556713346955520, 26433830208985721222400, 1406218428780691953635712

Offset: 0

Paul D. Hanna, Oct 30 2011

nonn

Compare to e.g.f. G(x) of A052802, which satisfies: G(x) = 1/(1 + log(1 - x*G(x))).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 144*x^4/4! + 1634*x^5/5! + ...
where log(1 + x*A(x)) equals
1 - 1/A(x) = x + x^2/2! + 5*x^3/3! + 38*x^4/4! + 404*x^5/5! + 5514*x^6/6! + ...

Cf. A030178, A052802.

a[n_] := Sum[ Binomial[n+k, n]*Sum[ (-1)^(j)*Binomial[k, j]*Sum[ (-1)^i*i!*Binomial[j, i]*StirlingS1[n, i], {i, 0, j}], {j, 0, k}], {k, 0, n}]/(n+1); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jun 24 2013, after Vladimir Kruchinin *)
CoefficientList[1/x*InverseSeries[Series[x-x*Log[1+x], {x, 0, 20}], x],x] * Range[0, 19]! (* Vaclav Kotesovec, Dec 28 2013 *)

a(n):=sum(binomial(n+k,n)*sum((-1)^(j)*binomial(k,j)*sum((-1)^i*i!*binomial(j,i)* stirling1(n,i),i,0,j),j,0,k),k,0,n)/(n+1); /* Vladimir Kruchinin, Feb 04 2012 */

{a(n)=n!*polcoeff((1/x)*serreverse(x-x*log(1+x +x*O(x^n))),n)}

{a(n)=n!*polcoeff(1/(1-log(1+x +x*O(x^n)))^(n+1)/(n+1),n)}

/* by Vladimir Kruchinin's formula: */
{a(n)=sum(k=0,n,binomial(n+k,n)*sum(j=0,k,(-1)^(j)*binomial(k,j)*sum(i=0,j,(-1)^i*i!*binomial(j,i)*stirling(n,i,1))))/(n+1)}

E.g.f. satisfies: A(x*(1 - log(1+x))) = 1/(1 - log(1+x)).
E.g.f.: A(x) = (1/x)*Series_Reversion[x - x*log(1+x)].
a(n) = n!*[x^n] 1/(1 - log(1+x))^(n+1)/(n+1).
a(n) = Sum_{k=0..n} (binomial(n+k,n) * Sum_{j=0..k} (-1)^(j)*binomial(k,j) * (Sum_{i=0..j} (-1)^i*i!*binomial(j,i)*Stirling1(n,i)))/(n+1). - Vladimir Kruchinin, Feb 04 2012
a(n) ~ n^(n-1) / ((1-c)*sqrt(1+c) * exp(n) * (1/c+c-2)^n), where c = LambertW(1). - Vaclav Kotesovec, Dec 28 2013
a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * Stirling1(n,k). - Seiichi Manyama, Nov 06 2023

A370994 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + x^2log(1-x)) ).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 3060, 23688, 191520, 9698400, 158548320, 2304973440, 100716073920, 2627516361600, 58513944513024, 2512156283683200, 89046056086041600, 2739316757454950400, 124170651534918297600, 5440968468533003212800, 215067442349096186572800

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A052802, A370993, A370995.

Cf. A351503, A370989.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x^2*log(1-x)))/x))

a(n) = sum(k=0, n\3, (n+k)!*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n+k)! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A377325 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x))/A(x).

Original entry on oeis.org

1, 1, 1, 5, 28, 244, 2566, 33438, 508544, 8926944, 176989488, 3917823216, 95719041408, 2559130965312, 74312569125744, 2329169772108528, 78371469374088960, 2817744760964392704, 107807187260426164992, 4373419962377871956736, 187507942522161269068800

Offset: 0

Seiichi Manyama, Oct 24 2024

nonn

Cf. A052802, A138013, A367159.

Cf. A365438, A377323.

a(n) = sum(k=0, (n+1)\2, (n-k)!/(n-2*k+1)!*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..floor((n+1)/2)} (n-k)!/(n-2*k+1)! * |Stirling1(n,k)|.

A370938 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + log(1-2x)/2) ).

Original entry on oeis.org

1, 1, 6, 68, 1152, 26144, 745952, 25678512, 1036151680, 47977039488, 2507929819392, 146106188393472, 9387670177320960, 659534185673994240, 50299364999073742848, 4138631751212820025344, 365438936930414973419520, 34469156894239754317332480

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A052802, A370939.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+log(1-2*x)/2))/x))

a(n) = sum(k=0, n, 2^(n-k)*(n+k)!*abs(stirling(n, k, 1)))/(n+1)!;

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^(n-k) * (n+k)! * |Stirling1(n,k)|.

A052819 E.g.f. equals the series reversion of x + x*log(1-x).

Original entry on oeis.org

0, 1, 2, 15, 188, 3300, 74484, 2054864, 66998448, 2520581400, 107472778320, 5121576763512, 269759385873504, 15561785854196400, 975788232119245440, 66080957140527828480, 4806533577745476290304, 373724762062131412853760

Offset: 0

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

			E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 188*x^4/4! + 3300*x^5/5! + ...
where A(x) = x - A(x)*log(1-A(x)).
The e.g.f. satisfies [from _Paul D. Hanna_, Jul 15 2012]:
(1) A(x) = x - x*log(1-x) + d/dx x^2*log(1-x)^2/2! - d^2/dx^2 x^3*log(1-x)^3/3! + d^3/dx^3 x^4*log(1-x)^4/4! + ...
(2) log(A(x)/x) = -log(1-x) + d/dx x*log(1-x)^2/2! - d^2/dx^2 x^2*log(1-x)^3/3! + d^3/dx^3 x^3*log(1-x)^4/4! + ...

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 784
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Cf. A052802. - Paul D. Hanna, Aug 28 2008

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

Flatten[{0,Table[Sum[k!*Abs[StirlingS1[n-1,k]]*Binomial[n+k-1,n-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 24 2013 *)
(1+r)*(2+r)/E/.FindRoot[(1+r)*(r+LambertW[-1,-E^(-r)*r]) == -r, {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n)/n, Vaclav Kotesovec, Sep 24 2013 *)

a(n):=(sum(k!*(-1)^(n+k-1)*stirling1(n-1,k)*binomial(n+k-1,n-1),k,0,n-1)); /* Vladimir Kruchinin, Feb 01 2012 */

a(n)=n!*polcoeff(serreverse(x+x*log(1-x +x*O(x^n))),n) \\ Paul D. Hanna, Aug 28 2008

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*(-log(1-x+x*O(x^n)))^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 15 2012

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*(-log(1-x+x*O(x^n)))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 15 2012

E.g.f. satisfies: A(x + x*log(1-x)) = x. - Paul D. Hanna, Aug 28 2008
E.g.f. A(x) satisfies [from Paul D. Hanna, Jul 15 2012]:
(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*(-log(1-x))^n/n!.
(2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*(-log(1-x))^n/n! ).
a(n) = Sum_{k=0..n-1} k!*(-1)^(n+k-1)*Stirling1(n-1,k)*binomial(n+k-1,n-1). - Vladimir Kruchinin, Feb 01 2012
Lim_{n->infinity} a(n)^(1/n)/n = (1+r)*(2+r)/exp(1) = 1.84542896220833..., where r = 0.794862961852611133... is the root of the equation (1+r)*(r+LambertW(-1,-r*exp(-r))) = -r. - Vaclav Kotesovec, Sep 24 2013

A367160 E.g.f. satisfies A(x) = 1 - A(x)^3 * log(1 - x*A(x)).

Original entry on oeis.org

1, 1, 9, 161, 4412, 164024, 7721898, 440550102, 29548655696, 2278884869640, 198709176600696, 19331290511231280, 2075887738522499664, 243905626745780976144, 31125204862136659763040, 4287017279890868817295728, 633888142969750426274770944

Offset: 0

Seiichi Manyama, Nov 07 2023

nonn

Cf. A052802, A138013, A367159.

a(n) = sum(k=0, n, (n+3*k)!/(n+2*k+1)!*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} (n+3*k)!/(n+2*k+1)! * |Stirling1(n,k)|.

cp's OEIS Frontend

A370993 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x*log(1-x)) ).

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A367139 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^3)).

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A367138 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^2)).

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A367159 E.g.f. satisfies A(x) = 1 - A(x)^2 * log(1 - x*A(x)).

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A198860 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x))).

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A370994 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^2*log(1-x)) ).

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A377325 E.g.f. satisfies A(x) = 1 - log(1 - x*A(x))/A(x).

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A370938 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + log(1-2*x)/2) ).

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A052819 E.g.f. equals the series reversion of x + x*log(1-x).

A370993 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + xlog(1-x)) ).

A370994 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + x^2log(1-x)) ).

A370938 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + log(1-2x)/2) ).