A220181 -id:A220181 - cp's OEIS frontend

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

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

Offset: 0

Vladeta Jovovic, Aug 31 2006

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

			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 + ...

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

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

a := n -> add(k^n*k!*combinat[stirling2](n,k),k=0..n); # Max Alekseyev, Feb 01 2007

Flatten[{1,Table[Sum[k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)

{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

{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

/* 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

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

More terms from Max Alekseyev, Feb 01 2007

A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.

Original entry on oeis.org

1, 2, 14, 230, 6902, 329462, 22934774, 2193664790, 276054834902, 44222780245622, 8787513806478134, 2121181056663291350, 611373265185174628502, 207391326125004608457782, 81791647413265571604175094, 37109390748309009878392597910, 19192672725746588045912535407702

Offset: 1

N. J. A. Sloane, R. K. Guy

nonn

a(n) is also the number of max-closed relations on an ordered n-element domain (see the paper by Jeavons and Cooper, 1995). - Don Knuth, Feb 12 2024

			1
1 + 1 = 2
1 + 9 + 4 = 14
1 + 49 + 144 + 36 = 230
1 + 225 + 2500 + 3600 + 576 = 6902
... - _Philippe Deléham_, May 30 2015

Lovasz, L. and Vesztergombi, K.; Restricted permutations and Stirling numbers. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 731-738, Colloq. Math. Soc. Janos Bolyai, 18, North-Holland, Amsterdam-New York, 1978.
K. Vesztergombi, Permutations with restriction of middle strength, Stud. Sci. Math. Hungar., 9 (1974), 181-185.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.
Peter G. Jeavons and Martin C. Cooper, Tractable constraints on ordered domains, Artificial Intelligence 79 (1995), 327-339.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.
H.-K. Kim et al., Poly-Bernoulli numbers and lonesum matrices, arXiv:1103.4884 [math.CO], 2011.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).

Main diagonal of array A099594.

Cf. A220181, A266695, A306209.

Table[Sum[((k-1)!)^2*StirlingS2[n,k]^2,{k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Jun 21 2013 *)

a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1)*(m-1)!*x^m/prod(k=1, m-1, 1+m*k*x+x*O(x^n))), n)) \\ Paul D. Hanna, Jan 05 2013
for(n=1,20,print1(a(n),", "))

Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)
a(n)=sum(k=1,n,(-1)^(n-k)*k^(n-1)*(k-1)!*Stirling2(n-1, k-1))
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 06 2013

a(n) = sum(k=1, n, (k-1)!^2*stirling(n,k,2)^2); \\ Michel Marcus, Jun 22 2018

E.g.f. (with offset 0): Sum((1-exp(-(m+1)*z))^m, m=0..oo)
O.g.f.: Sum_{n>=1} n^(n-1) * (n-1)! * x^n / Product_{k=1..n-1} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... . - Vaclav Kotesovec, Jun 21 2013
a(n) ~ 2*sqrt(Pi) * n^(2*n-3/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n-1)). - Vaclav Kotesovec, May 13 2014
a(n+1) = Sum_{k = 0..n} A163626(n,k)^2. - Philippe Deléham, May 30 2015
a(n) = A306209(2n-2,n-1). - Alois P. Heinz, Feb 01 2019
a(n) = A266695(2n-2). - Alois P. Heinz, Apr 17 2024

Entry revised by N. J. A. Sloane, Jul 05 2012

A229234 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - nkx).

Original entry on oeis.org

1, 1, 3, 19, 189, 2671, 50253, 1203679, 35548509, 1263153631, 52973381853, 2581493517439, 144317666200029, 9156299509121311, 653254398215833053, 51995430120141924799, 4585316010326597014749, 445304380297565009962591, 47368550666889620425580253, 5492643630110295899167573759

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 189*x^4 + 2671*x^5 + 50253*x^6 +...
where
A(x) = 1 + x/(1-x) + 2!*x^2/((1-2*1*x)*(1-2*2*x)) + 3!*x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4!*x^4/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 189*x^4/4! + 2671*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/2^2 + (exp(3*x)-1)^3/3^3 + (exp(4*x)-1)^4/4^4 + (exp(5*x)-1)^5/5^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..300

Cf. A229233, A220181, A108459, A122399.

Cf. A229258, A229259, A229260, A229261.

Flatten[{1,Table[Sum[k^(n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

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

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

{a(n)=sum(k=0, n, k^(n-k) * k! * stirling(n, k, 2))}
for(n=0,30,print1(a(n),", "))

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

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

A229260 O.g.f.: Sum_{n>=0} n! * n^(2n) x^n / Product_{k=1..n} (1 - n^2kx).

Original entry on oeis.org

1, 1, 33, 4759, 1812645, 1432421311, 2033196095973, 4707913008727279, 16598602853910799125, 84603008117292025844671, 598699398082553327852353413, 5694542805400507375406964870799, 70891082687197321771955383523878005, 1129717853570486718325946169950885995231

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

			O.g.f.: A(x) = 1 + x + 33*x^2 + 4759*x^3 + 1812645*x^4 + 1432421311*x^5 +...
where
A(x) = 1 + x/(1-x) + 2!*2^4*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^6*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^8*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 33*x^2/2! + 4759*x^3/3! + 1812645*x^4/4! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2 + (exp(9*x)-1)^3 + (exp(16*x)-1)^4 + (exp(25*x)-1)^5 + (exp(36*x)-1)^6 + (exp(49*x)-1)^7 +...

G. C. Greubel, Table of n, a(n) for n = 0..160

Cf. A229257, A229258, A229259, A229261, A229233, A229234, A220181, A122399.

Cf. A187755.

Flatten[{1,Table[Sum[k^(2*n) * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

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

{a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m),n)}
for(n=0,20,print1(a(n),", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=sum(k=0, n, k^(2*n) * k! * Stirling2(n, k))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} k^(2*n) * k! * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n.
a(n) ~ c * d^n * (n!)^3 / n, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.142680262107781025906560380273234930916319644... . - Vaclav Kotesovec, May 08 2014

A229261 O.g.f.: Sum_{n>=0} n^(2n) x^n / Product_{k=1..n} (1 - n^2kx).

Original entry on oeis.org

1, 1, 17, 922, 106695, 21742971, 6977367418, 3273755821827, 2129976884025085, 1846718792259030760, 2068516760060790309349, 2919795339100534415091143, 5088912154987483773753872912, 10766599670032172748225017763021, 27254500086981764567988714050736205

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

			O.g.f.: A(x) = 1 + x + 17*x^2 + 922*x^3 + 106695*x^4 + 21742971*x^5 +...
where
A(x) = 1 + x/(1-x) + 2^4*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3^6*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4^8*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 17*x^2/2! + 922*x^3/3! + 106695*x^4/4! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2! + (exp(9*x)-1)^3/3! + (exp(16*x)-1)^4/4! + (exp(25*x)-1)^5/5! + (exp(36*x)-1)^6/6! +...

Seiichi Manyama, Table of n, a(n) for n = 0..194

Cf. A229257, A229258, A229259, A229260, A229233, A229234, A220181, A122399.

Flatten[{1,Table[Sum[k^(2*n) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

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

{a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/m!),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=sum(k=0, n, k^(2*n) * stirling(n, k, 2))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} k^(2*n) * Stirling2(n, k).

E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n!.

A338040 E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j.

Original entry on oeis.org

1, 4, 132, 11140, 1763076, 449262724, 168055179012, 86720706877060, 59029852191779076, 51241585497612147844, 55245853646893977682692, 72423868722672448652558980, 113447698393867318106045295876, 209271794145089904620369489016964

Offset: 0

Vaclav Kotesovec, Oct 08 2020

nonn

In general, if k > 0 and e.g.f.: Sum_{j>=0} k^j * (exp(j*x) - 1)^j, then a(n) ~ c * (1 + k*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/k and c is a constant (dependent only on k).

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 [4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, 4, 6, 3, 0, 1, 0, ...], with an apparent period of 6. - Peter Bala, May 31 2022

Seiichi Manyama, Table of n, a(n) for n = 0..207

Cf. A122399, A195005, A195263, A195415, A220181, A221077, A221078, A224899, A245322.

Cf. A122400, A301581, A301582, A301583.

Flatten[{1, Table[Sum[4^j * j^n * j! * StirlingS2[n, j], {j, 0, n}], {n, 1, 20}]}]
nmax = 20; CoefficientList[Series[1 + Sum[4^j*(Exp[j*x] - 1)^j, {j, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 4^m*(exp(m*X)-1)^m), n)}

a(n) = Sum_{j=0..n} 4^j * j^n * j! * Stirling2(n,j).

a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.37483929689722634406486945426531890297038414869116425498643733178324...

A224899 E.g.f.: Sum_{n>=0} sinh(n*x)^n.

Original entry on oeis.org

1, 1, 8, 163, 6272, 389581, 35560448, 4479975823, 744707981312, 157897753198201, 41585725184933888, 13318468253704790683, 5097100004294081380352, 2297277197389011910783621, 1204339195916670860817072128, 726625952070893090583192860743

Offset: 0

Paul D. Hanna, Jul 24 2013

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, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, ...], with an apparent period of 6. Cf. A245322. - Peter Bala, May 29 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 163*x^3/3! + 6272*x^4/4! +...
where
A(x) = 1 + sinh(x) + sinh(2*x)^2 + sinh(3*x)^3 + sinh(4*x)^4 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..225

Cf. A122399, A249489, A245322, A220181, A221077, A221078, A198513, A220181, A249459, A195415, A245322, A338040.

Flatten[{1,Table[Sum[Sum[Binomial[k,j] * (-1)^j * k^n*(k-2*j)^n / 2^k,{j,0,k}],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 29 2014 *)
Join[{1},Rest[With[{nn=20},CoefficientList[Series[Sum[Sinh[n*x]^n,{n,nn}],{x,0,nn}],x] Range[0,nn]!]]] (* Harvey P. Dale, May 18 2018 *)

{a(n)=n!*polcoeff(sum(k=0, n, sinh(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: Sum_{n>=0} exp(-n^2*x) * (exp(2*n*x) - 1)^n / 2^n.

a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / (sqrt(3-2*log(2)) * 3^(n+1/2) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Oct 28 2014

A229233 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - nkx).

Original entry on oeis.org

1, 1, 2, 8, 48, 387, 4043, 52425, 819346, 15133184, 324769270, 7986143453, 222514878501, 6958782341565, 242274294115558, 9324382604206368, 394282071192289024, 18218582054356563951, 915480348188869318723, 49812603754178905560085, 2923492374797360684715882

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

Compare to an o.g.f. of Bell numbers (A000110): Sum_{n>=0} x^n/Product_{k=1..n} (1-k*x).

			O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 48*x^4 + 387*x^5 + 4043*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-2*1*x)*(1-2*2*x)) + x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + x^4/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 48*x^4/4! + 387*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/(2!*2^2) + (exp(3*x)-1)^3/(3!*3^3) + (exp(4*x)-1)^4/(4!*4^4) + (exp(5*x)-1)^5/(5!*5^5) +...

Seiichi Manyama, Table of n, a(n) for n = 0..332

Cf. A229234, A220181, A108459, A122399.

Cf. A229258, A229259, A229260, A229261.

Flatten[{1,Table[Sum[k^(n-k) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-m*k*x +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

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

{a(n)=sum(k=0, n, k^(n-k) * stirling(n, k, 2))}
for(n=0,30,print1(a(n),", "))

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

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

A320096 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.

Original entry on oeis.org

1, 1, 9, 212, 9418, 675014, 71092502, 10334690232, 1982433606264, 485065343565072, 147433546709109408, 54493722609862927632, 24069397682825072219040, 12520250948941157091235344, 7575515622713954399390221008, 5275250174853125498317783254528

Offset: 0

Vaclav Kotesovec, Oct 05 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..229

Cf. A007840, A220181, A320083.

Flatten[{1, Table[Sum[(-1)^(n-k)*StirlingS1[n, k]*k!*k^n, {k, 1, n}], {n, 1, 20}]}]
nmax = 20; CoefficientList[Series[1 + Sum[(-Log[1 - k*x])^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 04 2022 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^n*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 02 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k*x))^k))) \\ Seiichi Manyama, Feb 02 2022

a(n) ~ c * d^n * n^(2*n + 1/2), where
w = -LambertW(-1, -exp(-r)*r) = 1.1628296650659469964248518258036278907318113...
r = 0.8531304407911771560472963194514988627832723535823134189532... is the real root of the equation w = r + exp(-1/r)
d = exp(-1)*r*w*(w-r)^(r-1) = 0.433513333588184444899487502412976956849408575992...
c = 1.959633090979666812031505093625147349925787002426082...
E.g.f.: Sum_{k>=0} (-log(1 - k*x))^k. - Seiichi Manyama, Feb 02 2022

A204064 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + nx) / (1 + kx + n*x^2).

Original entry on oeis.org

1, 1, 2, 5, 14, 44, 152, 572, 2324, 10124, 47012, 231572, 1204964, 6599444, 37924292, 228033332, 1431128804, 9354072404, 63548071172, 447923400692, 3270361265444, 24696229801364, 192625876675652, 1549890430643252, 12849460733123684, 109647468132256724

Offset: 0

Paul D. Hanna, Jan 09 2013

nonn

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 + k*x + x^2) = (1+x^2)/(1-x).
Compare to the g.f. of A187741:
Sum_{n>=0} x^n * (1 + n*x)^n / (1 + x + n*x^2)^n  =  1/2 + (1+2*x)*Sum_{n>=0} (n+1)!*x^(2*n)/2.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 44*x^5 + 152*x^6 + 572*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+2*x)*(2+2*x)/((1+x+2*x^2)*(1+2*x+2*x^2)) + x^3*(1+3*x)*(2+3*x)*(3+3*x)/((1+x+3*x^2)*(1+2*x+3*x^2)*(1+3*x+3*x^2)) + x^4*(1+4*x)*(2+4*x)*(3+4*x)*(4+4*x)/((1+x+4*x^2)*(1+2*x+4*x^2)*(1+3*x+4*x^2)*(1+4*x+4*x^2)) +...
Also, we have the identity (cf. A229046):
A(x) = 1/2 + (1/2)*(1+x)/(1+x) + (2!/2)*x*(1+x)^2/((1+x)*(1+2*x)) + (3!/2)*x^2*(1+x)^3/((1+x)*(1+2*x)*(1+3*x)) + (4!/2)*x^3*(1+x)^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + (5!/2)*x^4*(1+x)^5/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) +...

Alois P. Heinz, Table of n, a(n) for n = 0..597

Cf. A229046, A204066, A187741, A187742, A220181.

b:= proc(n, k) option remember; `if`(n<1, 1, `if`(k>
      ceil(n/2), 0, add((k-j)*b(n-1-j, k-j), j=0..1)))
    end:
a:= n-> ceil(add(b(n+2, k), k=1..1+ceil(n/2))/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 26 2018

b[n_, k_] := b[n, k] = If[n < 1, 1, If[k > Ceiling[n/2], 0, Sum[(k - j) b[n - 1 - j, k - j], {j, 0, 1}]]];
a[n_] := Ceiling[Sum[b[n + 2, k], {k, 1, 1 + Ceiling[n/2]}]/2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

{a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1,m,(k+m*x)/(1+k*x+m*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=polcoeff( 1/2 + sum(m=1, n+1, m!/2*x^(m-1)*(1+x)^m/prod(k=1, m, 1+k*x +x*O(x^n))), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=if(n<0,0,if(n<1,1,(1/2)*sum(k=0, floor((n+1)/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k+1)))))} \\ Paul D. Hanna, Jul 13 2014
for(n=0, 30, print1(a(n), ", "))

G.f.: 1/2 + Sum_{n>=1} n!/2 * x^(n-1) * (1+x)^n / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Oct 27 2013
a(n) = A229046(n+1)/2 for n>0.
a(n) = (1/2)*Sum_{k=0..floor((n+1)/2)} Sum_{i=0..k} (-1)^i * C(k,i) * (k-i+1)^(n-k+1) for n>1. (See Paul Barry's formula in A105795). - Paul D. Hanna, Jul 13 2014

cp's OEIS Frontend

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

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A048163 a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2.

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A229234 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n*k*x).

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A229260 O.g.f.: Sum_{n>=0} n! * n^(2*n) * x^n / Product_{k=1..n} (1 - n^2*k*x).

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A229261 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / Product_{k=1..n} (1 - n^2*k*x).

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A338040 E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j.

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A229234 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - nkx).

A229260 O.g.f.: Sum_{n>=0} n! * n^(2n) x^n / Product_{k=1..n} (1 - n^2kx).

A229261 O.g.f.: Sum_{n>=0} n^(2n) x^n / Product_{k=1..n} (1 - n^2kx).

A229233 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - nkx).

A204064 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + nx) / (1 + kx + n*x^2).