A108459 -id:A108459 - cp's OEIS frontend

A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.

1, 2, 18, 314, 8434, 314362, 15278642, 928696442, 68509258098, 5995762219514, 611538502747826, 71656036268121978, 9532232740451770866, 1425414297318661354746, 237588200534263288095538, 43821269448954050939558522, 8887255081413035850889914994, 1970841722610600810208914571258, 475544555000142351430865220032434, 124299766720856839788225909600114042, 35056463298676734373530025799446104818

Offset: 0

Paul D. Hanna, Jun 28 2019

nonn

More generally, the following sums are equal:
Sum_{n>=0} (p + q^n)^n * r^n/n! =
Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = -1, r = 2.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = exp(x), p = -1, r = 2, m = 1.

			E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 314*x^3/3! + 8434*x^4/4! + 314362*x^5/5! + 15278642*x^6/6! + 928696442*x^7/7! + 68509258098*x^8/8! + 5995762219514*x^9/9! + 611538502747826*x^10/10! + ...
such that
A(x) = 1 + 2*(exp(x) - 1) + 2^2*(exp(2*x) - 1)^2/2! + 2^3*(exp(3*x) - 1)^3/3! + 2^4*(exp(4*x) - 1)^4/4! + 2^5*(exp(5*x) - 1)^5/5! + 2^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-2) + 2*exp(x)*exp(-2*exp(x)) + 2^2*exp(4*x)*exp(-2*exp(2*x))/2! + 2^3*exp(9*x)*exp(-2*exp(3*x))/3! + 2^4*exp(16*x)*exp(-2*exp(4*x))/4! + 2^5*exp(25*x)*exp(-2*exp(5*x))/5! + 2^6*exp(36*x)*exp(-2*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 2*x + 18*x^2 + 314*x^3 + 8434*x^4 + 314362*x^5 + 15278642*x^6 + 928696442*x^7 + 68509258098*x^8 + 5995762219514*x^9 + ...
such that
B(x) = 1 + 2*x/(1-x) + 2^2*2^2*x^2/((1-2*x)*(1-4*x)) + 2^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 2^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 2^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!  =  Sum_{n>=0} 2^n * exp(n^2*x) * exp( -2*exp(n*x) ) / n!.
(1) At x = -1, the following sums are equal
S1 = Sum_{n>=0} (-2)^n * (1 - exp(-n))^n / n!,
S1 = Sum_{n>=0} 2^n * exp(-n^2) * exp( -2*exp(-n) ) / n!,
where S1 = 0.51596189603321982013621912500044621350106513780391377129738...
(2) At x = -2, the following sums are equal
S2 = Sum_{n>=0} (-2)^n * (1 - exp(-2*n))^n / n!,
S2 = Sum_{n>=0} 2^n * exp(-2*n^2) * exp( -2*exp(-2*n) ) / n!,
where S2 = 0.34246794778612083304129071190905516612972983097016819355092...
(3) At x = -log(2), the following sums are equal
S3 = Sum_{n>=0} 2^(-n*(n-1)) * (2^n - 1)^n * (-1)^n / n!,
S3 = Sum_{n>=0} 2^(-n*(n-1)) * exp( -1/2^(n-1) ) / n!,
where S3 = 0.58106816860114387883649557314841837351794236167582918403231...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A108459, A326271, A326288.

Flatten[{1, Table[Sum[2^k * k^n * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 09 2019 *)

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

/* E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n! */
{a(n) = n! * polcoeff(sum(m=0, n, 2^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
for(n=0,30,print1(a(n),", "))

/* O.g.f.: Sum_{n>=0} 2^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 2^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.
E.g.f.: Sum_{n>=0} 2^n * exp(n^2*x) * exp( -2*exp(n*x) ) / n!.
O.g.f.: Sum_{n>=0} 2^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).
a(n) = Sum_{k=0..n} 2^k * k^n * Stirling2(n,k).

A326271 E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n!.

Original entry on oeis.org

1, 3, 39, 948, 34869, 1757163, 114320118, 9226773993, 897658726215, 103005144933870, 13705015429716807, 2085418048857405375, 358813807291982519184, 69146346672687725451039, 14803634157756603606592167, 3496440993213535696041009924, 905508769623362527769907535857, 255762146004426658313683324505247, 78413243604482526944814375388910526, 25983968388767783226046397856822603645

Offset: 0

Paul D. Hanna, Jun 28 2019

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = -1, r = 3.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = exp(x), p = -1, r = 3, m = 1.

			E.g.f.: A(x) = 1 + 3*x + 39*x^2/2! + 948*x^3/3! + 34869*x^4/4! + 1757163*x^5/5! + 114320118*x^6/6! + 9226773993*x^7/7! + 897658726215*x^8/8! + 103005144933870*x^9/9! + ...
such that
A(x) = 1 + 3*(exp(x) - 1) + 3^2*(exp(2*x) - 1)^2/2! + 3^3*(exp(3*x) - 1)^3/3! + 3^4*(exp(4*x) - 1)^4/4! + 3^5*(exp(5*x) - 1)^5/5! + 3^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-3) + 3*exp(x)*exp(-3*exp(x)) + 3^2*exp(4*x)*exp(-3*exp(2*x))/2! + 3^3*exp(9*x)*exp(-3*exp(3*x))/3! + 3^4*exp(16*x)*exp(-3*exp(4*x))/4! + 3^5*exp(25*x)*exp(-3*exp(5*x))/5! + 3^6*exp(36*x)*exp(-3*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 3*x + 39*x^2 + 948*x^3 + 34869*x^4 + 1757163*x^5 + 114320118*x^6 + 9226773993*x^7 + 897658726215*x^8 + 103005144933870*x^9 + ...
such that
B(x) = 1 + 3*x/(1-x) + 3^2*2^2*x^2/((1-2*x)*(1-4*x)) + 3^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 3^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 3^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A108459, A326270, A326288.

{a(n) = sum(k=0, n, 3^k * k^n * stirling(n, k, 2) )}
for(n=0, 30, print1(a(n), ", "))

/* E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n! */
{a(n) = n! * polcoeff(sum(m=0, n, 3^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
for(n=0, 30, print1(a(n), ", "))

/* O.g.f.: Sum_{n>=0} 3^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 3^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n!.
E.g.f.: Sum_{n>=0} 3^n * exp(n^2*x) * exp( -3*exp(n*x) ) / n!.
O.g.f.: Sum_{n>=0} 3^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).
a(n) = Sum_{k=0..n} 3^k * k^n * Stirling2(n,k).

A326288 E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n!.

Original entry on oeis.org

1, 4, 68, 2116, 98436, 6217924, 503491204, 50282169284, 6023071906180, 847321700204740, 137695169475601540, 25505309294030757316, 5326002105122774427524, 1242268006104279981404868, 321107726934189274515747460, 91359880704866957348006879172, 28441686041231472428045000672644, 9637951929231839144943126955386052, 3538621024404268912313596289954242692, 1401869934089183216934147248975602680260

Offset: 0

Paul D. Hanna, Jun 28 2019

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = exp(x) with p = -1, r = 4.
In general, let F(x) be a formal power series in x such that F(0)=1, then
Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =
Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
here, F(x) = exp(x), q = exp(x), p = -1, r = 4, m = 1.

			E.g.f.: A(x) = 1 + 4*x + 68*x^2/2! + 2116*x^3/3! + 98436*x^4/4! + 6217924*x^5/5! + 503491204*x^6/6! + 50282169284*x^7/7! + 6023071906180*x^8/8! + 847321700204740*x^9/9! + ...
such that
A(x) = 1 + 4*(exp(x) - 1) + 4^2*(exp(2*x) - 1)^2/2! + 4^3*(exp(3*x) - 1)^3/3! + 4^4*(exp(4*x) - 1)^4/4! + 4^5*(exp(5*x) - 1)^5/5! + 4^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-4) + 4*exp(x)*exp(-4*exp(x)) + 4^2*exp(4*x)*exp(-4*exp(2*x))/2! + 4^3*exp(9*x)*exp(-4*exp(3*x))/3! + 4^4*exp(16*x)*exp(-4*exp(4*x))/4! + 4^5*exp(25*x)*exp(-4*exp(5*x))/5! + 4^6*exp(36*x)*exp(-4*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 4*x + 68*x^2 + 2116*x^3 + 98436*x^4 + 6217924*x^5 + 503491204*x^6 + 50282169284*x^7 + 6023071906180*x^8 + 847321700204740*x^9 + ...
such that
B(x) = 1 + 4*x/(1-x) + 4^2*2^2*x^2/((1-2*x)*(1-4*x)) + 4^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 4^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 4^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A108459, A326270, A326271.

{a(n) = sum(k=0, n, 4^k * k^n * stirling(n, k, 2) )}
for(n=0, 30, print1(a(n), ", "))

/* E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n! */
{a(n) = n! * polcoeff(sum(m=0, n, 4^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
for(n=0, 30, print1(a(n), ", "))

/* O.g.f.: Sum_{n>=0} 4^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 4^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n!.
E.g.f.: Sum_{n>=0} 4^n * exp(n^2*x) * exp( -4*exp(n*x) ) / n!.
O.g.f.: Sum_{n>=0} 4^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).
a(n) = Sum_{k=0..n} 4^k * k^n * Stirling2(n,k).

A350722 a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 33, 4567, 1652493, 1235777551, 1656820330173, 3619858882041487, 12034209740498292093, 57813156798714532953391, 385490564193781368103929213, 3454086424032897924417605526607, 40500898779980258599522326286912893

Offset: 0

Seiichi Manyama, Feb 03 2022

nonn

Cf. A122399, A195005, A195263, A338040.

Cf. A108459, A350721, A351117.

a[0] = 1; a[n_] := Sum[k! * k^(k+n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 03 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k)))

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

a(n) ~ exp(exp(-2)/2) * n! * n^(2*n). - Vaclav Kotesovec, Feb 04 2022

A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 5, 10, 15, 0, 1, 9, 22, 37, 52, 0, 1, 17, 52, 99, 151, 203, 0, 1, 33, 130, 283, 471, 674, 877, 0, 1, 65, 340, 855, 1561, 2386, 3263, 4140, 0, 1, 129, 922, 2707, 5451, 8930, 12867, 17007, 21147, 0, 1, 257, 2572, 8919, 19921, 35098, 53411, 73681, 94828, 115975

Offset: 1

Christian G. Bower, Jun 03 2005; Emeric Deutsch, Nov 14 2006

Another way to obtain this sequence (with offset 0): Form the infinite array U(n,k) = number of labeled partitions of (n,k) into pairs (i,j), for n >= 0, k >= 0 and read it by antidiagonals. In other words, U(n,k) = number of partitions of n black objects labeled 1..n and k white objects labeled 1..k. Each block must have at least one white object.

Then T(n,k)=U(n+k,k+1). Thus the two versions are related like "multichoose" to "choose". - Augustine O. Munagi, Jul 16 2007

			Triangle T(n,k) starts:
  1;
  0,1;
  0,1,2;
  0,1,3,5;
  0,1,5,10,15;
T(5,3)=5 because we have 1245|3, 145|2|3, 14|25|3, 15|24|3 and 1|245|3.
The arrays U(n,k) starts:
   1  0  0   0   0 ...
   1  1  1   1   1 ...
   2  3  5   9  17 ...
   5 10 22  52 130 ...
  15 37 99 283 855 ...

Alois P. Heinz, Rows n = 1..141, flattened

Row sums of T(n, k) yield A124496(n, 1).
Cf. A108461.
Columns of U(n, k): A000110, A005493, A033452.
Rows of U(n, k): A000007, A000012, A000051.
Main diagonal: A108459.

T[n_, k_] := Sum[If[n == k, 1, i^(n-k)]*StirlingS2[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2024, after Vladeta Jovovic *)

T(n,1)=0 for n>=2; T(n,2)=1 for n>=2; T(n,3)=1+2^(n-3) for n>=3; T(n,n)=B(n-1), T(n,n-1)=B(n-1)-B(n-2), where B(q) are the Bell numbers (A000110).
Double e.g.f.: exp(exp(x)*(exp(y)-1)).
U(n,k) = Sum_{i=0..k} i^(n-k)*Stirling2(k,i). - Vladeta Jovovic, Jul 12 2007

Edited by N. J. A. Sloane, May 22 2008, at the suggestion of Vladeta Jovovic. This entry is a composite of two entries submitted independently by Christian G. Bower and Emeric Deutsch, with additional comments from Augustine O. Munagi.

A128943 a(n) = Sum_{k=0..n} (-1)^(n-k)k^nStirling1(n,k).

Original entry on oeis.org

1, 1, 5, 53, 924, 23494, 810872, 36194514, 2017775680, 136829739216, 11055586913832, 1046742607228152, 114550470343202880, 14323855468574034720, 2026669209500208676608, 321743057984308274403024, 56892680614922936544276480, 11133427829583046292676364800, 2397458024796587973818060252160

Offset: 0

Vladeta Jovovic, May 09 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..43

Cf. A108459.

with(combinat): a:=n->sum((-1)^(n-k)*k^n*stirling1(n,k),k=0..n): seq(a(n),n=0..18); # Emeric Deutsch, May 18 2007

Table[Sum[Abs[StirlingS1[n+1,k+1]]StirlingS2[n,k]k!,{k,0,n}],{n,0,100}] (* Emanuele Munarini, Jul 04 2011 *)
nmax = 20; CoefficientList[Series[1 + Sum[(-Log[1 - k*x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 04 2022 *)

makelist(sum(abs(stirling1(n+1,k+1))*stirling2(n,k)*k!,k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */

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

a(n) = Sum_{k=0..n} abs(Stirling1(n+1,k+1))*Stirling2(n,k)*k!. - Emanuele Munarini, Jul 04 2011

More terms from Emeric Deutsch, May 18 2007

A198513 E.g.f.: Sum_{n>=0} sinh(n*x)^n/n!.

Original entry on oeis.org

1, 1, 4, 28, 320, 5556, 129600, 3756936, 132872192, 5679982288, 286769980416, 16732506817280, 1115928688967680, 84383735744758464, 7163164003950936064, 676619301019539271040, 70674282825174467215360, 8117559039240651749888256

Offset: 0

Paul D. Hanna, Oct 26 2011

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 320*x^4/4! + 5556*x^5/5! +...
where
A(x) = 1 + sinh(x) + sinh(2*x)^2/2! + sinh(3*x)^3/3! + sinh(4*x)^4/4! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A108459.

{a(n)=local(A=sum(m=0,n,sinh(m*x+x*O(x^n))^m/m!));n!*polcoeff(A,n)}

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

A367820 Number of partitions of [2n] that have at most one block contained in [n].

Original entry on oeis.org

1, 2, 13, 153, 2744, 68303, 2224417, 90995838, 4538437039, 269755223485, 18766884323562, 1506040068195721, 137740473851280141, 14212098473767962472, 1640078704487165930485, 210103319793655159244093, 29684467774817808296383256, 4598958815992575305097910699

Offset: 0

Alois P. Heinz, Dec 01 2023

nonn

			a(2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3.

Alois P. Heinz, Table of n, a(n) for n = 0..288

Cf. A008277, A048993, A011655, A108459, A113547, A362925.

b:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> add(coeff(b(n), x, j)*(j+1)^n, j=0..n):
seq(a(n), n=0..21);

A367820[n_]:=Sum[StirlingS2[n,j](j+1)^n,{j,0,n}];
Array[A367820,25,0] (* Paolo Xausa, Dec 04 2023 *)

a(n) = A113547(2n+1,n+1) = A362925(2n,n).
a(n) = Sum_{j=0..n} (j+1)^n * Stirling2(n,j).
a(n) mod 2 = A011655(n+2).

A351181 a(n) = Sum_{k=0..n} k^(k+n) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 17, 826, 79107, 12553011, 2979141058, 988163147091, 436562014218313, 247800100563125728, 175732698005376526429, 152264214647249387402567, 158273183995563848011907696, 194391589002961482387840145341

Offset: 0

Seiichi Manyama, Feb 04 2022

nonn

Cf. A108459, A229233, A229261, A282190, A308490.

Cf. A350722, A351180.

a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)

a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 2));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k/k!)))

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

a(n) ~ c * r^(2*n) * (1 + exp(1 + 1/r))^n * n^(2*n) / exp(2*n), where r = 0.942405403803582963024019065398882138211529545249588032669864757847... is the root of the equation r*(1 + exp(-1 - 1/r)) * LambertW(-exp(-1/r)/r) = -1 and c = 0.94346979328254581112250921799629823027437848684764713214690470878402... - Vaclav Kotesovec, Feb 18 2022

A356772 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} ( x^n + x*A(x) )^n / n!.

Original entry on oeis.org

1, 2, 5, 34, 329, 3716, 55777, 1010206, 21187049, 511352272, 13929248861, 422450642054, 14129873671069, 516664310959720, 20503766568423881, 877759284120870526, 40321132468408643153, 1978363648482263649728, 103262474042895179595061, 5713315282015940379009862

Offset: 0

Paul D. Hanna, Aug 27 2022

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = x with p = x*A(x), r = 1.

			E.g.f.: A(x) = 1 + 2*x + 5*x^2/2! + 34*x^3/3! + 329*x^4/4! + 3716*x^5/5! + 55777*x^6/6! + 1010206*x^7/7! + 21187049*x^8/8! + 511352272*x^9/9! + 13929248861*x^10/10! + ...
where
A(x) = 1 + (x + x*A(x)) + (x^2 + x*A(x))^2/2! + (x^3 + x*A(x))^3/3! + (x^4 + x*A(x))^4/4! + (x^5 + x*A(x))^5/5! + ... + (x^n + x*A(x))^n/n! + ...
also
A(x) = exp(x*A(x)) + x*exp(x^2*A(x)) + x^4*exp(x^3*A(x))/2! + x^9*exp(x^4*A(x))/3! + x^16*exp(x^5*A(x))/4! + x^25*exp(x^6*A(x))/5! + ... +  + x^(n^2)*exp(x^(n+1)*A(x))/n! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 5*x^2/2! + 28*x^3/3! + 269*x^4/4! + 3356*x^5/5! + 50257*x^6/6! + 915076*x^7/7! + 19427753*x^8/8! + 471310984*x^9/9! + 12892968701*x^10/10! + ...
log(A(x)) = 2*x + x^2/2! + 20*x^3/3! + 126*x^4/4! + 1314*x^5/5! + 20460*x^6/6! + 347906*x^7/7! + 7181944*x^8/8! + 170606106*x^9/9! + 4577504760*x^10/10! + ...
SPECIFIC VALUES.
A(x = 1/4) = 1.8854569251645435475372616427080...
A(x = 0.3) = 2.4910587821818158559566392662113...
A(x = 1/3) diverges.

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A356773, A108459, A326090, A326091, A326261, A326009.

/* A(x) = Sum_{n>=0} ( x^n + x*A(x) )^n / n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (x^m + x*A +x*O(x^n))^m/m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

/* A(x) = Sum_{n>=0} x^(n^2) * exp(x^(n+1)*A(x))/n! */
{a(n) = my(A=1); for(i=1,n, A = sum(m=0,sqrtint(n), x^(m^2) * exp( x^(m+1)*A +x*O(x^n)) / m! )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = Sum_{n>=0} ( x^n + x*A(x) )^n / n!.
(2) A(x) = Sum_{n>=0} x^(n^2) * exp( x^(n+1) * A(x) ) / n!.
a(n) ~ c * d^n * n! / n^(3/2), where d = 3.14614463757985697... and c = 1.454175198420213... - Vaclav Kotesovec, Jul 03 2025

cp's OEIS Frontend

A326270 E.g.f.: Sum_{n>=0} 2^n * (exp(n*x) - 1)^n / n!.

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A326271 E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n!.

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A326288 E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n!.

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A350722 a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling2(n,k).

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A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

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A128943 a(n) = Sum_{k=0..n} (-1)^(n-k)*k^n*Stirling1(n,k).

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A198513 E.g.f.: Sum_{n>=0} sinh(n*x)^n/n!.

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A128943 a(n) = Sum_{k=0..n} (-1)^(n-k)k^nStirling1(n,k).