A217900 -id:A217900 - cp's OEIS frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

4, 0, 6, 3, 7, 5, 7, 3, 9, 9, 5, 9, 9, 5, 9, 9, 0, 7, 6, 7, 6, 9, 5, 8, 1, 2, 4, 1, 2, 4, 8, 3, 9, 7, 5, 8, 2, 1, 0, 9, 9, 7, 5, 7, 5, 1, 8, 1, 1, 4, 0, 6, 3, 5, 0, 0, 0, 4, 9, 5, 4, 8, 8, 3, 0, 3, 9, 1, 5, 0, 1, 5, 1, 8, 3, 8, 1, 2, 0, 4, 9, 7, 6, 7, 2, 5, 0, 0, 7, 2, 3, 3, 8, 1, 5, 5, 9, 2, 8, 5, 8, 2, 9, 3, 8

Offset: 0

José María Grau Ribas, Jun 17 2013

			-0.4063757399599599076769581241248397582109975751811406350004954883....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.

RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)

solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017

Equals -2*A106533.

Equals LambertW(-2*exp(-2)).

A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(nx)^n/n! exp(-nxA(n*x)).

Original entry on oeis.org

1, 1, 2, 9, 63, 659, 9833, 206961, 6133990, 256650268, 15213478000, 1281205909177, 153588353066135, 26245044813624300, 6399076697684238375, 2227912079081482302977, 1108302173165578509079527, 788171767077184315422131588, 801638519723021288783092512047

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identities:
(1) Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).
(2) Sum_{n>=0} n^n * x^n * C(x)^n/n! * exp(-n*x*C(x)) = C(x), where C(x) = 1 + x*C(x)^2 is the o.g.f. of the Catalan numbers (A000108).

			O.g.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 63*x^4 + 659*x^5 + 9833*x^6 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^2*x^2*A(2*x)^2/2!*exp(-2*x*A(2*x)) + 3^3*x^3*A(3*x)^3/3!*exp(-3*x*A(3*x)) + 4^4*x^4*A(4*x)^4/4!*exp(-4*x*A(4*x)) + 5^5*x^5*A(5*x)^5/5!*exp(-5*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A218673, A218674, A218675, A218676, A218681.
Cf. A217900, A218670, A218667, A218668, A218669, A134055.
Cf. A193363, A221409, A221410, A221411, A221412, A221413.
Cf. A209276, A209277.

a[n_] := Module[{A}, A[x_] = 1 + x; For[i = 1, i <= n, i++, A[x_] = Sum[If[k == 0, 1, k^k] x^k A[k x]^k/k! Exp[-k x A[k x] + x O[x]^i] // Normal, {k, 0, n}]]; Coefficient[ A[x], x, n]];
a /@ Range[0, 18] (* Jean-François Alcover, Sep 29 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A,x,k*x)^k/k!*exp(-k*x*subst(A,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A217913 O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3x) x^n / n!.

Original entry on oeis.org

1, 1, 31, 3025, 611501, 210766920, 110687251039, 82310957214948, 82318282158320505, 106563273280541795575, 173373343599189364594756, 346289681454731077633095526, 833091176987705031151553054843, 2376102520162485084539597049185710

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

			O.g.f.: A(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + ... + Stirling2(3*n, n)*x^n + ...
where
A(x) = 1 + 1^3*x*exp(-1^3*x) + 2^6*exp(-2^3*x)*x^2/2! + 3^9*exp(-3^3*x)*x^3/3! + 4^12*exp(-4^3*x)*x^4/4! + 5^15*exp(-5^3*x)*x^5/5! + ...
simplifies to a power series in x with integer coefficients.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A219228, A007820, A217914, A217915, A217900, A008277.

Table[StirlingS2[3*n,n],{n,0,20}] (* Vaclav Kotesovec, Feb 28 2013 *)

makelist(stirling2(3*n, n), n, 0, 13); /* Martin Ettl, Oct 15 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^3)^k*exp(-k^3*x +x*O(x^n))*x^k/k!),n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^3)^k*x^k/(1+k^3*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(3*n, n)}
for(n=0,20,print1(a(n),", "))

a(n) = Stirling2(3*n, n).
a(n) = [x^(3*n)] (3*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(2*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^3)^k*x^k / (1 + k^3*x)^(k+1).
a(n) ~ 9^n*exp(n*(c+1))*n^(2*n)/((c+3)^(2*n)*sqrt(2*Pi*(c+1)*n)), where c = -0.1785606278779211... = LambertW(-3/exp(3)) = A226750. - Vaclav Kotesovec, Feb 28 2013

A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

Original entry on oeis.org

1, 2, 4, 1, 4, 32, 38, 12, 1, 8, 208, 652, 576, 188, 24, 1, 16, 1280, 9080, 16944, 12052, 3840, 580, 40, 1, 32, 7744, 116656, 412800, 540080, 322848, 98292, 16000, 1390, 60, 1, 64, 46592, 1446368, 9196992, 20447056, 20453376, 10564304, 3047520, 511392, 50400

Offset: 1

N. J. A. Sloane, Dec 21 2002

A generalization of the Stirling2 numbers S_{1,1} from A008277.
The g.f. for column k=2*K is (x^K)*pe(K,x)*d(k,x) and for k=2*K+1 it is (x^K)*po(K,x)*2*(K+1)*K*d(k,x), K>= 1, with d(k,x) := 1/product(1-p*(p-1)*x,p=2..k) and the row polynomials pe(n,x) := sum(A089275(n,m)*x^m,m=0..n-1) and po(n,x) := sum(A089276(n,m)*x^m,m=0..n-1). - Wolfdieter Lang, Nov 07 2003
The formula for the k-th column sequence is given in A089511.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_2 (the disjoint union of n copies of the complete graph K_2). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k | 2    3    4    5    6   7   8
= = = = = = = = = = = = = = = = = =
  1 | 1
  2 | 2    4    1
  3 | 4   32   38   12    1
  4 | 8  208  652  576  188  24   1
...
Graph coloring interpretation of T(2,3) = 4: The graph 2K_2 is 2 copies of K_2, the complete graph on 2 vertices:
o---o  o---o
a   b  c   d
The four 3-colorings of 2K_2 are ac|b|d, ad|b|c, bc|a|d and bd|a|c. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Steve Butler, Fan Chung, Jay Cummings, R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
Leonard Carlitz, On Arrays of Numbers, Am. J. Math., 54,4 (1932) 739-752. [Eqs. (3) and (4) with lambda = 0, mu = 2, a_{n,k-1} = a(n, k).- _Wolfdieter Lang_, Jan 30 2020 ]
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

Row sums give A020556. Triangle S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{3, 2} = A078740, S_{3, 3} = A078741. A090214 (S_{4,4}).
The column sequences are A000079(n-1)(powers of 2), 4*A016129(n-2), A089271, 12*A089272, A089273, etc.
Main diagonal is A217900.
Cf. A071951 (Legendre-Stirling triangle).
Cf. A122193, A055203.

# Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A078739 := proc(n,k) local r;
add((-1)^(n-r)*binomial(n,r)*combinat[stirling2](n+r,k),r=0..n) end:
# Displays the truncated triangle from the definition:
seq(print(seq(A078739(n,k),k=2..2*n)),n=1..6); # Peter Luschny, Mar 25 2011

t[n_, k_] := Sum[(-1)^(n-r)*Binomial[n, r]*StirlingS2[n+r, k], {r, 0, n}]; Table[t[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Peter Luschny *)

a(n, k) = sum(binomial(k-2+p, p)*A008279(2, p)*a(n-1, k-2+p), p=0..2) if 2 <= k <= 2*n for n>=1, a(1, 2)=1; else 0. Here A008279(2, p) gives the third row (k=2) of the augmented falling factorial triangle: [1, 2, 2] for p=0, 1, 2. From eq.(21) with r=2 of the Blasiak et al. paper.
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*A008279(p, 2)^n, p=2..k) for 2 <= k <= 2*n, n>=1. From eq.(19) with r=2 of the Blasiak et al. paper.
a(n, k) = sum(A071951(n, j)*A089503(j, 2*j-k+1), j=ceiling(k/2)..min(n, k-1)), 1<=n, 2<=k<=2n; relation to Legendre-Stirling triangle. Wolfdieter Lang, Dec 01 2003
a(n, k) = A122193(n,k)*2^n/k! - Peter Luschny, Mar 25 2011
E^n = sum_{k=2}^(2n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^2d^2/dx^2.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*sum {k = 0..inf} (k*(k-1))^n*x^k/k!. - Peter Bala, Aug 15 2013

More terms from Wolfdieter Lang, Nov 07 2003

A218670 O.g.f.: Sum_{n>=0} n^n * (1+nx)^n x^n/n! * exp(-nx(1+n*x)).

Original entry on oeis.org

1, 1, 2, 7, 26, 116, 556, 2927, 16388, 97666, 612136, 4023878, 27579410, 196537134, 1451102836, 11074811191, 87160086800, 706055915318, 5876662642720, 50182337830986, 439036984440316, 3930618736372336, 35970734643745496, 336153100655220126, 3205000520319374116

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare the o.g.f. to the curious identity:

1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).

			O.g.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 116*x^5 + 556*x^6 + 2927*x^7 +...
where
A(x) = 1 + (1+x)*x*exp(-x*(1+x)) + 2^2*(1+2*x)^2*x^2/2!*exp(-2*x*(1+2*x)) + 3^3*(1+3*x)^3*x^3/3!*exp(-3*x*(1+3*x)) + 4^4*(1+4*x)^4*x^4/4!*exp(-4*x*(1+4*x)) + 5^5*(1+5*x)^5*x^5/5!*exp(-5*x*(1+5*x)) +...
simplifies to a power series in x with integer coefficients.

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A218677, A218678, A218679, A218667, A218668, A218669, A134055, A218671, A217900.

{a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^k*x^k/k!*exp(-k*x*(1+k*x)+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A218667 O.g.f.: Sum_{n>=0} 1/(1-nx)^n x^n/n! * exp(-x/(1-n*x)).

Original entry on oeis.org

1, 0, 1, 1, 4, 13, 46, 181, 778, 3585, 17566, 91171, 499324, 2873839, 17317743, 108933098, 713481122, 4855161425, 34257461754, 250177938679, 1887886966690, 14699340919293, 117933068390123, 973776266303732, 8265721830953558, 72052688932613079, 644393453082317301

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare g.f. to the curious identity:

1/(1+x^2) = Sum_{n>=0} (1-n*x)^n * x^n/n! * exp(-x*(1-n*x)).

			O.g.f.: A(x) = 1 + x^2 + x^3 + 4*x^4 + 13*x^5 + 46*x^6 + 181*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-2*x)^2/2!*exp(-x/(1-2*x)) + x^3/(1-3*x)^3/3!*exp(-x/(1-3*x)) + x^4/(1-4*x)^4/4!*exp(-x/(1-4*x)) + x^5/(1-5*x)^5/5!*exp(-x/(1-5*x)) + x^6/(1-6*x)^6/6!*exp(-x/(1-6*x)) +...
simplifies to a power series in x with integer coefficients.

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A245111, A245110, A218668, A218669, A218670, A185040, A217900.

{a(n)=local(A=1+x,X=x+x*O(x^n));A=sum(k=0,n,1/(1-k*X)^k*x^k/k!*exp(-X/(1-k*X)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

/* From a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(-1, k-1) */
{Stirling2(n, k) = sum(j=0, k, (-1)^(k+j) * binomial(k, j) * j^n) / k!}
{a(n)=if(n==0, 1, sum(k=1, n, Stirling2(n-k, k) * binomial(n-1, k-1)))}
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=1..n} Stirling2(n-k, k) * C(n-1, k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jul 30 2014

Antidiagonal sums of Triangle A245111.

A218300 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(2x)cosh(2x).

Original entry on oeis.org

1, 2, 12, 104, 1216, 18112, 329600, 7108096, 177549312, 5046554624, 160947232768, 5694342479872, 221410157133824, 9387011838312448, 431051678297358336, 21316106766591721472, 1129526392342026649600, 63855305138514241257472, 3836490516381680506241024

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 104*x^3/3! + 1216*x^4/4! + 18112*x^5/5! +...
where
A(x) = cosh(2*x) + 2*3^0*cosh(3*x)*x + 2*4^1*cosh(4*x)*x^2/2! + 2*5^2*cosh(5*x)*x^3/3! + 2*6^3*cosh(6*x)*x^4/4! + 2*7^4*cosh(7*x)*x^5/5! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A201595, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310, A217900.

Cf. A202357.

nmin = 0; nmax = 18; sol = {a[0] -> 1}; nsol = Length[sol];
Do[A[x_] = Sum[a[k] x^k/k!, {k, 0, n}] /. sol; eq = CoefficientList[ A[x/(Exp[x] Cosh[x])] - Exp[2x] Cosh[2x] + O[x]^(n+1), x][[nsol+1;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol+1, nmax}];
a /@ Range[nmin, nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(2*R)*cosh(2*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,2*(k+2)^(k-1)*cosh((k+2)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} 2*(n+2)^(n-1) * cosh((n+2)*x) * x^n/n!.
E.g.f.: A(x) = 1 + Sum_{n>=0} 2*(n+2)^(n-1) * sinh((n+2)*x) * x^n/n!.
a(n) ~ c * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/LambertW(exp(-1))^2 = 14.5815783688217906961670551786416446... . - Vaclav Kotesovec, Jul 13 2014, updated Jun 10 2019
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 2*x - 2*LambertW(-x * exp(x)) ).
a(n) = Sum_{k=0..n} (k+2)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + Sum_{k>=0} (k+2)^(k-1) * x^k/(1 - (k+2)*x)^(k+1). (End)

A217914 O.g.f.: Sum_{n>=0} (n^4)^n * exp(-n^4x) x^n / n!.

Original entry on oeis.org

1, 1, 127, 86526, 171798901, 749206090500, 6090236036084530, 82892803728383735268, 1751346256720122175776157, 54294340536065700496358447625, 2364684125291482936353925428946680, 139762001313639974628848043262243505970, 10897986831117690497797320098390628446479030

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

			O.g.f.: A(x) = 1 + x + 127*x^2 + 86526*x^3 + 171798901*x^4 +...+ Stirling2(4*n,n)*x^n + ...
where
A(x) = 1 + 1^4*x*exp(-1^4*x) + 2^8*exp(-2^4*x)*x^2/2! + 3^12*exp(-3^4*x)*x^3/3! + 4^16*exp(-4^4*x)*x^4/4! + 5^20*exp(-5^4*x)*x^5/5! + ...
is a power series in x with integer coefficients.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A007820, A217913, A217915, A217900, A008277.

Table[StirlingS2[4*n,n],{n,0,20}] (* Vaclav Kotesovec, May 23 2013 *)

makelist(stirling2(4*n, n), n, 0, 12); /* Martin Ettl, Oct 15 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^4)^k*exp(-k^4*x +x*O(x^n))*x^k/k!),n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^4)^k*x^k/(1+k^4*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(3*n))), 3*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(4*n, n)}
for(n=0,12,print1(a(n),", "))

a(n) = Stirling2(4*n, n).
a(n) = [x^(4*n)] (4*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(3*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^4)^k*x^k / (1 + k^4*x)^(k+1).
a(n) ~ 2^(8*n)*n^(3*n)/(sqrt(2*Pi*n*(1-c))*c^n*exp(3*n)*(4-c)^(3*n)), where c = -LambertW(-4/exp(4)) = 0.07930960512711... - Vaclav Kotesovec, May 23 2013

A217899 O.g.f.: Sum_{n>=1} (n^2)^(n-1) * exp(-n^2x) x^n / n!.

Original entry on oeis.org

1, 1, 6, 65, 1050, 22827, 627396, 20912320, 820784250, 37112163803, 1900842429486, 108823356051137, 6888836057922000, 477898618396288260, 36060660300744309600, 2940812098256837097720, 257780560811305783833450, 24171700822696604400643035, 2414448376056191692970387250

Offset: 1

Paul D. Hanna, Oct 14 2012

nonn

For n>1, a(n) is the number of set partitions of [2*n-2] into n blocks, i.e., Stirling2(2*n-2, n). E.g., a(3) = 6: [12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34]. - Yuchun Ji, Jan 12 2021

			O.g.f.: A(x) = x + x^2 + 6*x^3 + 65*x^4 + 1050*x^5 + 22827*x^6 + 627396*x^7 + ... where A(x) = 1^0*x*exp(-1*x) + 2^2*exp(-2^2*x)*x^2/2! + 3^4*exp(-3^2*x)*x^3/3! + 4^6*exp(-4^2*x)*x^4/4! + 5^8*exp(-5^2*x)*x^5/5! + ... simplifies to a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A217900, A217901, A217902, A217903, A217904, A217905.

a[n_] := Sum[ Binomial[2*n - 3, j]*StirlingS2[2*n - j - 3, n-1], {j, 0, n-2}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 14 2013, after Vladimir Kruchinin *)

{a(n)=polcoeff(sum(m=1,n,(m^2)^(m-1)*x^m*exp(-m^2*x+x*O(x^n))/m!),n)}

{a(n)=1/n!*sum(k=1,n, (-1)^(n-k)*binomial(n,k)*(k^2)^(n-1))}

{a(n)=polcoeff(x+x^2/prod(k=1, n, 1-k*x +x*O(x^n)), n)}

{a(n)=polcoeff(x+x^2*(1+x)^(2*n-3)/prod(k=0, n-1, 1-k*x +x*O(x^n)), n)}
for(n=1,20,print1(a(n),", "))

a(n) = (1/n!) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * (k^2)^(n-1).
a(n) = [x^n] x + x^2/Product_{k=1..n} (1-k*x).
a(n) = [x^n] x + x^2*(1+x)^(2*n-3) / Product_{k=1..n-1} (1-k*x).
a(n) = Sum_{j=0..n-1} binomial(2*n-1,j)*Stirling2(2*n-j-1,n). - Vladimir Kruchinin, Jun 14 2013
a(n) ~ 2^(2*n-5/2) * n^(n-5/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n *(2-c)^(n-2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 20 2014

A217915 O.g.f.: Sum_{n>=1} (n^5)^n * exp(-n^5x) x^n / n!.

Original entry on oeis.org

1, 1, 511, 2375101, 45232115901, 2436684974110751, 299310102746948685757, 72786959006434393367186463, 31712979422428631132831124895809, 22982258052528294182955639980819773510, 26154716515862881292012777396577993781727011

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

			O.g.f.: A(x) = 1 + x + 511*x^2 + 2375101*x^3 + 45232115901*x^4 +...+ Stirling2(5*n, n)*x^n +...
where
A(x) = 1 + 1^5*x*exp(-1^5*x) + 2^10*exp(-2^5*x)*x^2/2! + 3^15*exp(-3^5*x)*x^3/3! + 4^20*exp(-4^5*x)*x^4/4! + 5^25*exp(-5^5*x)*x^5/5! +...
is a power series in x with integer coefficients.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A007820, A217913, A217914, A217900, A008277.

Table[StirlingS2[5*n,n],{n,0,20}] (* Vaclav Kotesovec, May 23 2013 *)

makelist(stirling2(5*n, n), n, 0, 10); /* Martin Ettl, Oct 15 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^5)^k*exp(-k^5*x +x*O(x^n))*x^k/k!),n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^5)^k*x^k/(1+k^5*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(4*n))), 4*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(5*n, n)}
for(n=0,12,print1(a(n),", "))

a(n) = Stirling2(5*n, n).
a(n) = [x^(5*n)] (5*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(4*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^5)^k*x^k / (1 + k^5*x)^(k+1).
a(n) ~ n^(4*n)*5^(5*n) / (sqrt(2*Pi*n*(1-c)) * exp(4*n) * (5-c)^(4*n) * c^n), where c = -LambertW(-5/exp(5)) = 0.0348857682557... - Vaclav Kotesovec, May 23 2013

cp's OEIS Frontend

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

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A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n*x)^n/n! * exp(-n*x*A(n*x)).

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A217913 O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3*x) * x^n / n!.

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Maxima

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A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

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Maple

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A218670 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^n * x^n/n! * exp(-n*x*(1+n*x)).

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A218667 O.g.f.: Sum_{n>=0} 1/(1-n*x)^n * x^n/n! * exp(-x/(1-n*x)).

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A218300 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(2*x)*cosh(2*x).

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A218672 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(nx)^n/n! exp(-nxA(n*x)).

A217913 O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3x) x^n / n!.

A218670 O.g.f.: Sum_{n>=0} n^n * (1+nx)^n x^n/n! * exp(-nx(1+n*x)).

A218667 O.g.f.: Sum_{n>=0} 1/(1-nx)^n x^n/n! * exp(-x/(1-n*x)).

A218300 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(2x)cosh(2x).

A217914 O.g.f.: Sum_{n>=0} (n^4)^n * exp(-n^4x) x^n / n!.

A217899 O.g.f.: Sum_{n>=1} (n^2)^(n-1) * exp(-n^2x) x^n / n!.

A217915 O.g.f.: Sum_{n>=1} (n^5)^n * exp(-n^5x) x^n / n!.