cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A122399 a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).

Original entry on oeis.org

1, 1, 9, 211, 9285, 658171, 68504709, 9837380491, 1863598406805, 450247033371451, 135111441590583909, 49300373690091496171, 21495577955682021043125, 11037123350952586270549531, 6591700149366720366704735109
Offset: 0

Views

Author

Vladeta Jovovic, Aug 31 2006

Keywords

Comments

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 2, 1, 3, 3, 0, 1, 2, 1, 3, 3, 0, ...], with an apparent period of 6. Cf. A338040. - Peter Bala, May 31 2022

Examples

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + 9285*x^4/4! + 658171*x^5/5! + ...
such that
A(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2 + (exp(3*x)-1)^3 + (exp(4*x)-1)^4 + ...
The e.g.f. is also given by the series:
A(x) = 1/2 + exp(x)/(1+exp(x))^2 + exp(4*x)/(1+exp(2*x))^3 + exp(9*x)/(1+exp(3*x))^4 + exp(16*x)/(1+exp(4*x))^5 + exp(25*x)/(1+exp(5*x))^6 + ...
or, equivalently,
A(x) = 1/2 + exp(-x)/(1+exp(-x))^2 + exp(-2*x)/(1+exp(-2*x))^3 + exp(-3*x)/(1+exp(-3*x))^4 + exp(-4*x)/(1+exp(-4*x))^5 + exp(-5*x)/(1+exp(-5*x))^6 + ...

Links

Crossrefs

Cf. A195415, A122400, A220181, A224899, A245322, A320083, A338040.

Programs

  • Maple
    a := n -> add(k^n*k!*combinat[stirling2](n,k),k=0..n); # Max Alekseyev, Feb 01 2007
  • Mathematica
    Flatten[{1,Table[Sum[k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)
  • PARI
    {a(n)=polcoeff(sum(m=0, n, m^m*m!*x^m/prod(k=1, m, 1-m*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013
  • PARI
    {a(n)=n!*polcoeff(sum(k=0, n, (exp(k*x +x*O(x^n)) - 1)^k), n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Oct 26 2014
  • PARI
    /* From e.g.f. infinite series: */
\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n^2*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}
for(n=0, #A-1, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 30 2014

Formula

E.g.f.: Sum_{n >= 0} (exp(n*x) - 1)^n. - Vladeta Jovovic, Sep 03 2006
E.g.f.: Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x))^(n+1). - Paul D. Hanna, Oct 26 2014
E.g.f.: Sum_{n>=0} exp(-n*x) / (1 + exp(-n*x))^(n+1). - Paul D. Hanna, Oct 30 2014
O.g.f.: Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = ((1+exp(1/r))*r^2)/exp(1) = A317855/exp(1) = 1.162899527477400818845..., where r = 0.87370243323966833... is the root of the equation 1/(1+exp(-1/r)) = -r*LambertW(-exp(-1/r)/r). - Vaclav Kotesovec, Jun 21 2013
a(n) ~ c * A317855^n * (n!)^2 / sqrt(n), where c = 0.327628285569869481442286492410507030710253054522608... - Vaclav Kotesovec, Aug 09 2018
Let A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + ... denote the e.g.f. of the sequence. Let F(x) denote the series reversion of A(x) - 1 = x - 9*x^2/2 + 16*x^3/3 - 205*x^4/4 - 2714*x^5/5 - .... Then both dF/dx = 1 - 9*x + 16*x^2 - 205*x^3 - 2714*x^4 - ... and exp(F(x)) = 1 + x - 4*x^2 + x^3 - 38*x^4 - 606*x^5 - ... have integer coefficients. Note that 1 + series reversion(exp(F(x)) - 1) is the o.g.f. for A122400. - Peter Bala, Aug 09 2022

Extensions

More terms from Max Alekseyev, Feb 01 2007

A122400 Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.

Original entry on oeis.org

1, 1, 4, 31, 338, 4769, 82467, 1687989, 39905269, 1069863695, 32071995198, 1062991989013, 38596477083550, 1523554760656205, 64961391010251904, 2975343608212835855, 145687881987604377815, 7594435556630244257213
Offset: 0

Views

Author

Vladeta Jovovic, Aug 31 2006

Keywords

Links

Crossrefs

Cf. A104602, A220353, A301581, A301582, A301583, A301584.

Programs

  • Maple
    A122399 := proc(n) option remember ; add( combinat[stirling2](n,k)*k^n*k!,k=0..n) ; end: A122400 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122399(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122400(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    max = 17; CoefficientList[ Series[ 1 + Sum[ ((1 + x)^n - 1)^n, {n, 1, max}], {x, 0, max}], x] (* Jean-François Alcover, Mar 26 2013, after Vladeta Jovovic *)

Formula

a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A122399(k).
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n. - Vladeta Jovovic, Sep 03 2006
G.f.: Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1). - Paul D. Hanna, Mar 23 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.2796968489586733500739737080739303725411427162653658... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from R. J. Mathar, May 18 2007

A303056 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.

Original entry on oeis.org

1, 1, 1, 8, 89, 1326, 24247, 521764, 12867985, 357229785, 11017306489, 373675921093, 13825260663882, 554216064798423, 23934356706763264, 1108017262467214486, 54747529760516714323, 2876096694574711401525, 160092696678371426933342, 9413031424290635395882462, 583000844360279565483710624
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2018

Keywords

Comments

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 1, p = -A(x), q = (1+x). - Paul D. Hanna, Jun 22 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 89*x^4 + 1326*x^5 + 24247*x^6 + 521764*x^7 + 12867985*x^8 + 357229785*x^9 + 11017306489*x^10 + ...
such that
1 = 1  +  ((1+x) - A(x))  +  ((1+x)^2 - A(x))^2  +  ((1+x)^3 - A(x))^3  +  ((1+x)^4 - A(x))^4  +  ((1+x)^5 - A(x))^5  +  ((1+x)^6 - A(x))^6  +  ((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1+x)/(1 + (1+x)*A(x))^2  +  (1+x)^4/(1 + (1+x)^2*A(x))^3  +  (1+x)^9/(1 + (1+x)^3*A(x))^4  +  (1+x)^16/(1 + (1+x)^4*A(x))^5  +  (1+x)^25/(1 + (1+x)^5*A(x))^6  +  (1+x)^36/(1 + (1+x)^6*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = x + x^2/2 + 22*x^3/3 + 325*x^4/4 + 6186*x^5/5 + 137380*x^6/6 + 3478651*x^7/7 + 98674253*x^8/8 + 3096911434*x^9/9 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1/2, it may be evaluated formally.
Let t = A(-1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).
Also,
A(r) = 1/2 at r = -0.54683649902292991492196620520872286547799291909992048564578...
where
(1) 1 = Sum_{n>=0} ( (1+r)^n - 1/2 )^n.
(2) 1 = Sum_{n>=0} (1+r)^(-n) / ( 1/(1+r)^n + 1/2 )^(n+1).

Links

Crossrefs

Cf. A304642, A304639, A303926.
Cf. A321602, A321603, A321604, A321605.
Cf. A326282, A326283, A326284.
Cf. A337755, A337756, A337757.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ((1+x)^m - Ser(A))^m ) )[#A] );A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654... and c = 0.11739505492506... - Vaclav Kotesovec, Sep 26 2020

A244585 E.g.f.: Sum_{n>=1} (exp(n*x) - 1)^n / n.

Original entry on oeis.org

1, 5, 79, 2621, 149071, 12954365, 1596620719, 264914218301, 56934521042191, 15385666763366525, 5106110041462786159, 2041611328770984737981, 967972254733121945653711, 536962084044317668770841085, 344546100916295014902350596399
Offset: 1

Views

Author

Paul D. Hanna, Aug 21 2014

Keywords

Comments

Compare to: Sum_{n>=1} (1 - exp(-n*x))^n / n, the e.g.f. of A092552.

Examples

			E.g.f.: A(x) = x + 5*x^2/2! + 79*x^3/3! + 2621*x^4/4! + 149071*x^5/5! +...
where
A(x) = (exp(x)-1) + (exp(2*x)-1)^2/2 + (exp(3*x)-1)^3/3 + (exp(4*x)-1)^4/4 + (exp(5*x)-1)^5/5 + (exp(6*x)-1)^6/6 + (exp(7*x)-1)^7/7 +...
Exponentiation yields:
exp(A(x)) = 1 + x + 6*x^2/2! + 95*x^3/3! + 3043*x^4/4! + 167342*x^5/5! +...+ A243802(n)*x^n/n! +...
The O.G.F. begins:
F(x) = x + 5*x^2 + 79*x^3 + 2621*x^4 + 149071*x^5 + 12954365*x^6 +...
where
F(x) = x/(1-x) + 2*2!*x^2/((1-2*x)*(1-4*x)) + 3^2*3!*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 4^3*4!*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 5^4*5!*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) +...

Links

Crossrefs

Cf. A092552, A243802.

Programs

  • PARI
    {a(n) = n!*polcoeff( sum(m=1,n+1, (exp(m*x +x*O(x^n)) - 1)^m / m), n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1) * m! * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n))}
for(n=0, 20, print1(a(n), ", "))

Formula

O.g.f.: Sum_{n>=1} n^(n-1) * n! * x^n / Product_{k=1..n} (1 - n*k*x).
a(n) ~ c * d^n * (n!)^2 / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491..., r = 0.873702433239668330496568304720719298... is the root of the equation exp(1/r)/r + (1+exp(1/r)) * LambertW(-exp(-1/r)/r) = 0, and c = 0.37498840921734807101035131780130551... . - Vaclav Kotesovec, Aug 21 2014

A304639 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.

Original entry on oeis.org

1, 1, 2, 11, 117, 1735, 31853, 689043, 17079221, 476238926, 14742680162, 501584454703, 18605089712174, 747393133162471, 32332767332220442, 1498961537925543920, 74153115616699819304, 3899494667155151052688, 217246028175467702590241, 12783023090792392539557926, 792236994094236725330142276, 51585659784100723438219893047, 3520987513029712770759434038820
Offset: 0

Views

Author

Paul D. Hanna, May 16 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 117*x^4 + 1735*x^5 + 31853*x^6 + 689043*x^7 + 17079221*x^8 + 476238926*x^9 + 14742680162*x^10 + 501584454703*x^11 + ...
is such that
1 = 1 + (1/(1-x) - A(x)) + (1/(1-x)^2 - A(x))^2  + (1/(1-x)^3 - A(x))^3 + (1/(1-x)^4 - A(x))^4 + (1/(1-x)^5 - A(x))^5 + (1/(1-x)^6 - A(x))^6 + (1/(1-x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  (1-x)/((1-x) + A(x))^2  +  (1-x)^2/((1-x)^2 + A(x))^3  +  (1-x)^3/((1-x)^3  +  A(x))^4 + (1-x)^4/((1-x)^4 + A(x))^5  +  (1-x)^5/((1-x)^5 + A(x))^6  +  (1-x)^6/((1-x)^6 + A(x))^7 + ...
PARTICULAR VALUES.
Although the power series A(x) diverges at x = -1, it may be evaluated formally.
Let t = A(-1) = 0.5452189736359494312349502450349441069576127988881794567242641...
then t satisfies
(1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
(2) 1 = Sum_{n>=0} ( 1 - 2^n*t )^n / 2^(n^2).
(3) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).

Links

Crossrefs

Cf. A326262, A326263, A326264, A326265.
Cf. A303056.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (1/(1-x +x^2*O(x^n))^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/(1-x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} ( 1 - (1-x)^n*A(x) )^n / (1-x)^(n^2).
(3) 1 = Sum_{n>=0} (1-x)^n / ( (1-x)^n + A(x) )^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.16107844724485... - Vaclav Kotesovec, Oct 14 2020

A122418 a(n) = Sum_{k=0..n} (k-1)^n*k!*Stirling2(n,k).

Original entry on oeis.org

1, 0, 2, 54, 2534, 186030, 19794662, 2885980734, 552803552534, 134687987183790, 40686498089484422, 14925683377452413214, 6536580413039406774134, 3368723388994026165415950, 2018248855531992511720945382, 1390953089533285777007059354494, 1092714503596231472933813958469334
Offset: 0

Views

Author

Vladeta Jovovic, Sep 03 2006

Keywords

Links

Crossrefs

Cf. A122419, A122420, A122399.

Programs

  • Maple
    A122418 := proc(n) sum((k-1)^n*k!*combinat[stirling2](n,k),k=0..n) ; end; for n from 0 to 16 do print(A122418(n)) ; od ; # R. J. Mathar, Feb 10 2007
  • Mathematica
    a[n_] := Sum[ (k-1)^n*k!*StirlingS2[n, k], {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 26 2013 *)
  • PARI
    for(n=0,50, print1(sum(k=0,n, (k-1)^n*k!*stirling(n,k,2)), ", ")) \\ G. C. Greubel, Nov 15 2017

Formula

E.g.f.: Sum((exp((n-1)*x)-1)^n, n=0..infinity).
a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.10430562057820038909699083625848223918044424242153125547162600916636313858475... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from R. J. Mathar, Feb 10 2007

A301584 G.f.: Sum_{n>=0} ((1+x)^(2*n) - 1)^n.

Original entry on oeis.org

1, 2, 17, 264, 5784, 163610, 5667551, 232280480, 10991951114, 589780778314, 35379149504709, 2346218124687516, 170439977706143335, 13459938431949414118, 1148107512505151099653, 105194122765096703619248, 10303686044959088279454117, 1074408525677705370497704526, 118828297870115694372235974855, 13893778686151373846512389392672, 1712370237144948501135060958863978
Offset: 0

Views

Author

Paul D. Hanna, Mar 24 2018

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 17*x^2 + 264*x^3 + 5784*x^4 + 163610*x^5 + 5667551*x^6 + 232280480*x^7 + 10991951114*x^8 + 589780778314*x^9 + ...
such that
A(x) = 1 + ((1+x)^2-1) + ((1+x)^4-1)^2 + ((1+x)^6-1)^3 + ((1+x)^8-1)^4 + ((1+x)^10-1)^5 + ((1+x)^12-1)^6 + ((1+x)^14-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^2/(1 + (1+x)^2)^2 + (1+x)^8/(1 + (1+x)^4)^3 + (1+x)^18/(1 + (1+x)^6)^4 + (1+x)^32/(1 + (1+x)^8)^5 + (1+x)^50/(1 + (1+x)^10)^6 + ...

Links

Crossrefs

Cf. A122400, A301585, A301586.

Programs

  • PARI
    {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, ((1+x +o)^(2*m) - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} (1+x)^(2*n^2) /(1 + (1+x)^(2*n))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 2*A317855 = 6.3221773077308576276603444051762649834527655483771126832545564150753941184386... and c = 0.302715376391132275494451399946850989516917... - Vaclav Kotesovec, Aug 09 2018

A122419 Number of labeled digraphs with n arcs and with no vertex of indegree 0.

Original entry on oeis.org

1, 0, 1, 8, 93, 1354, 23900, 496244, 11855700, 320428318, 9667220397, 322072882348, 11744421711587, 465270864839688, 19899234175413257, 913836170567749048, 44849438199960187278, 2342666125012348876152
Offset: 0

Views

Author

Vladeta Jovovic, Sep 03 2006

Keywords

Crossrefs

Cf. A122420, A122400.

Programs

  • Maple
    A122418 := proc(n) option remember ; add( combinat[stirling2](n,k)*(k-1)^n*k!,k=0..n) ; end: A122419 := proc(n) option remember ; add( combinat[stirling1](n,k)*A122418(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122419(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    nmax=20; CoefficientList[Series[Sum[((1+x)^(n-1)-1)^n, {n, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 06 2014 *)

Formula

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A122418(k).
G.f.: Sum_{n>=0} ((1+x)^(n-1) - 1)^n.
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.08904589343883135100956914504938... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from R. J. Mathar, May 18 2007

A122420 Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.

Original entry on oeis.org

1, 0, 1, 10, 120, 1778, 31685, 661940, 15882128, 430607370, 13022755068, 434697574538, 15875944361864, 629756003982336, 26963278837704185, 1239382820431888898, 60875147436141987437, 3181961834442383306068
Offset: 0

Views

Author

Vladeta Jovovic, Sep 03 2006

Keywords

Crossrefs

Cf. A104209.

Programs

  • Maple
    A122418 := proc(n) option remember ; add( combinat[stirling2](n,k)*(k-1)^n*k!,k=0..n) ; end: A122420 := proc(n) option remember ; add( abs(combinat[stirling1](n,k))*A122418(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122420(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    Table[1/n!*Sum[Abs[StirlingS1[n,k]]*Sum[(m-1)^k*m!*StirlingS2[k,m],{m,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)

Formula

a(n) = (1/n!)*Sum_{k=0..n} |Stirling1(n,k)|*A122418(k). G.f.: A(x/(1-x)) where A(x) is g.f. for A122419.
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.1221803955695846906452721220983425... . - Vaclav Kotesovec, May 07 2014

Extensions

More terms from R. J. Mathar, May 18 2007

A243802 E.g.f.: exp( Sum_{n>=1} (exp(n*x) - 1)^n / n ).

Original entry on oeis.org

1, 1, 6, 95, 3043, 167342, 14175447, 1715544861, 280986929888, 59828264507385, 16056622678756319, 5300955907062294008, 2110872493413444115109, 997542435957462115205773, 551887323312314977683048334, 353334615697796170374209624907, 259179558930246734075836153918127
Offset: 0

Views

Author

Paul D. Hanna, Aug 21 2014

Keywords

Comments

Compare to: exp( Sum_{n>=1} (exp(x) - 1)^n/n ) = 1/(2-exp(x)), the e.g.f. of Fubini numbers (A000670).

Examples

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 95*x^3/3! + 3043*x^4/4! + 167342*x^5/5! +...

Links

Crossrefs

Cf. A244585, A244437.

Programs

  • PARI
    {a(n) = n!*polcoeff( exp( sum(m=1,n+1, (exp(m*x +x*O(x^n)) - 1)^m / m) ), n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ c * d^n * (n!)^2 / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491..., r = 0.873702433239668330496568304720719298... is the root of the equation exp(1/r)/r + (1+exp(1/r)) * LambertW(-exp(-1/r)/r) = 0, and c = 0.37498840921734807101035131780130551... . - Vaclav Kotesovec, Aug 21 2014
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