A216689 -id:A216689 - cp's OEIS frontend

A216688 Expansion of e.g.f. exp( x * exp(x^2) ).

1, 1, 1, 7, 25, 121, 841, 4831, 40657, 325585, 2913841, 29910871, 301088041, 3532945417, 41595396025, 531109561711, 7197739614241, 100211165640481, 1507837436365537, 23123578483200295, 376697477235716281, 6348741961892933401, 111057167658053740201, 2032230051717594032767

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216689 (e.g.f. exp(x*exp(x)^2)).

Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x^2]], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x^2) )))
/* Joerg Arndt, Sep 14 2012 */

a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ Seiichi Manyama, Aug 18 2022

a(n) = n!*Sum_{m=floor((n+1)/2)..n} (2*m-n)^(n-m)/((2*m-n)!*(n-m)!). - Vladimir Kruchinin, Mar 09 2013
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (r^n * exp((2*r^2*n)/(1+2*r^2)) * sqrt(3+2*r^2 - 2/(1 + 2*r^2))), where r is the root of the equation r*exp(r^2)*(1+2*r^2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(2^(1/3)*n^(2/3)/3))) *  sqrt(2/(3*LambertW(2^(1/3)*n^(2/3)/3))).
(End)

A216507 E.g.f. exp( x^2 * exp(x) ).

Original entry on oeis.org

1, 0, 2, 6, 24, 140, 870, 5922, 45416, 381096, 3442410, 33382910, 345803172, 3801763836, 44156760830, 539962736250, 6929042527920, 93032248209872, 1303556965679826, 19018807375195638, 288341417011487420, 4534168069704168420, 73829219253218066022, 1242905562198878544626

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..516 (terms 0..200 from Vincenzo Librandi)
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292978.
Cf. A216688 (e.g.f. exp(x*exp(x^2))), A216689 (e.g.f. exp(x*exp(x)^2)).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

With[{nn = 25}, CoefficientList[Series[Exp[x^2 Exp[x]], {x, 0, nn}],
   x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x^2 * exp(x) )))
/* Joerg Arndt, Sep 14 2012 */

From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(n*(1+r)/(2+r)) * r^n * sqrt((1+r)*(4+r)/(2+r))), where r is the root of the equation r^2*(2+r)*exp(r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
(End)
a(n) = Sum_{k = 0..n/2} C(n,2*k) * ((2*k)!/k!) * k^(n-2*k). - David Einstein, Oct 30 2016

A240165 E.g.f.: exp( x(1 + exp(2x)) ).

Original entry on oeis.org

1, 2, 8, 44, 288, 2192, 18976, 182912, 1934848, 22231808, 275203584, 3645178880, 51370694656, 766634946560, 12066538676224, 199607631945728, 3459736006950912, 62662715180515328, 1183139425871331328, 23237689444403511296, 473852525131782946816, 10014501808427774246912

Offset: 0

Paul D. Hanna, Aug 02 2014

nonn

			E.g.f.: E(x) = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 288*x^4/4! + 2192*x^5/5! +...
where E(x) = exp(x) * exp(x*exp(2*x)).
O.g.f.: A(x) = 1 + 2*x + 8*x^2 + 44*x^3 + 288*x^4 + 2192*x^5 +...
where
A(x) = 1/(1-x) + x/(1-3*x)^2 + x^2/(1-5*x)^3 + x^3/(1-7*x)^4 + x^4/(1-9*x)^5 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A080108, A216689.

Table[Sum[Binomial[n,k] *(2*k+1)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x(1+Exp[2x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 17 2016 *)

{a(n)=local(A=1);A=exp( x*(1 + exp(2*x +x*O(x^n))) );n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

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

{a(n)=sum(k=0,n, binomial(n,k) * (2*k+1)^(n-k) )}
for(n=0,30,print1(a(n),", "))

O.g.f.: Sum_{n>=0} x^n / (1 - (2*n+1)*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (2*k+1)^(n-k) for n>=0.
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp((1+exp(2*r))*r - n) * n^(n+1/2) / (r^n * sqrt(r + exp(2*r)*r*(1 + 6*r + 4*r^2))), where r is the root of the equation r*(1 + exp(2*r) + 2*r*exp(2*r)) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
(End)

A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266

Offset: 0

Vaclav Kotesovec, Aug 06 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292892.
Cf. A000110, A052506, A216507, A060311.
Cf. A216688, A216689, A080108, A000248.

CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

A245834 E.g.f.: exp( x(1 + exp(3x)) ).

Original entry on oeis.org

1, 2, 10, 71, 592, 5777, 64792, 814025, 11264176, 169871633, 2768582104, 48412950929, 902831609368, 17865749820089, 373564063839376, 8223263706957713, 189960800250512608, 4591950749700004385, 115866075506169417256, 3044877330738661504625, 83169542349597382767496, 2356949307613191494567561

Offset: 0

Paul D. Hanna, Aug 02 2014

nonn

			E.g.f.: E(x) = 1 + 2*x + 10*x^2/2! + 71*x^3/3! + 592*x^4/4! + 5777*x^5/5! +...
where E(x) = exp(x) * exp(x*exp(3*x)).
O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 71*x^3 + 592*x^4 + 5777*x^5 + 64792*x^6 +...
where
A(x) = 1/(1-x) + x/(1-4*x)^2 + x^2/(1-7*x)^3 + x^3/(1-10*x)^4 + x^4/(1-13*x)^5 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A080108, A216689, A240165, A245835.

Table[Sum[Binomial[n,k] *(3*k+1)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x(1+Exp[3x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 09 2019 *)

{a(n)=local(A=1);A=exp( x*(1 + exp(3*x +x*O(x^n))) );n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - (3*k+1)*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

{a(n)=sum(k=0,n,(3*k+1)^(n-k)*binomial(n,k))}
for(n=0,30,print1(a(n),", "))

O.g.f.: Sum_{n>=0} x^n / (1 - (3*n+1)*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (3*k+1)^(n-k) for n>=0.
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp((1+exp(3*r))*r - n) * n^(n+1/2) / (r^n * sqrt(r + exp(3*r)*r* (1+9*r*(1+r)))), where r is the root of the equation r*(1 + exp(3*r) + 3*r*exp(3*r)) = n.
(a(n)/n!)^(1/n) ~ 3*exp(1/(2*LambertW(sqrt(3*n)/2))) / (2*LambertW(sqrt(3*n)/2)).
(End)

A116071 Triangle T, read by rows, equal to Pascal's triangle to the matrix power of Pascal's triangle, so that T = C^C, where C(n,k) = binomial(n,k) and T(n,k) = A000248(n-k)*C(n,k).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 41, 40, 18, 4, 1, 196, 205, 100, 30, 5, 1, 1057, 1176, 615, 200, 45, 6, 1, 6322, 7399, 4116, 1435, 350, 63, 7, 1, 41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1, 293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1

Offset: 0

Paul D. Hanna, Feb 03 2006

Column 0 = A000248 (Number of forests with n nodes and height at most 1).
Column 1 = A052512 (Number of labeled trees of height 2).
Row sums = A080108 (Sum_{k=1..n} k^(n-k) * C(n-1,k-1)).
Central terms = A116072(n) = (n+1) * A000108(n) * A000248(n).
From Peter Bala, Sep 13 2012: (Start)
For commuting lower unitriangular matrix A and lower triangular matrix B we define A raised to the matrix power B, denoted by A^B, to be the lower unitriangular matrix Exp(B*Log(A)). Here Exp denotes the matrix exponential defined by the power series
  Exp(A) = 1 + A + A^2/2! + A^3/3! + ...
and the matrix logarithm Log(A) is defined by the series
  Log(A) = (A-1) - 1/2*(A-1)^2/2 + 1/3*(A-1)^3 - ....
Let A = [f(x),x] and B = [g(x),x] be exponential Riordan arrays in the Appell subgroup and suppose f(0) = 1. Then A and B commute and A^B is the exponential Riordan array [exp(g(x)*log(f(x))),x], also belonging to the Appell group. In the present case we are taking A = B = [exp(x),x], equal to the Pascal triangle A007318.
For any lower unitriangular matrix A (with, say, rational entries) the infinite tower of powers A^(A^(A^(...))) is well-defined (and also has rational entries). An example is given in the Formula section. (End)

			E.g.f.: E(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2/2!
  + (10 + 9*y + 3*y^2 + y^3)*x^3/3!
  + (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4/4!
  + (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5/5! +...
where E(x,y) = exp(x*y) * exp(x*exp(x)).
O.g.f.: A(x,y) = 1 + (1 + y)*x + (3 + 2*y + y^2)*x^2
  + (10 + 9*y + 3*y^2 + y^3)*x^3
  + (41 + 40*y + 18*y^2 + 4*y^3 + y^4)*x^4
  + (196 + 205*y + 100*y^2 + 30*y^3 + 5*y^4 + y^5)*x^5 +...
where
A(x,y) = 1/(1-x*y) + x/(1-x*(y+1))^2 + x^2/(1-x*(y+2))^3 + x^3/(1-x*(y+3))^4 + x^4/(1-x*(y+4))^5 + x^5/(1-x*(y+5))^6 + x^6/(1-x*(y+6))^7 + x^7/(1-x*(y+7))^8 +...
Triangle begins:
  1;
  1, 1;
  3, 2, 1;
  10, 9, 3, 1;
  41, 40, 18, 4, 1;
  196, 205, 100, 30, 5, 1;
  1057, 1176, 615, 200, 45, 6, 1;
  6322, 7399, 4116, 1435, 350, 63, 7, 1;
  41393, 50576, 29596, 10976, 2870, 560, 84, 8, 1;
  293608, 372537, 227592, 88788, 24696, 5166, 840, 108, 9, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080 (rows 0..45 of flattened triangle).

Cf. A000248, A052512, A080108, A116072, A215652.

Cf. A080108, A216689, A240165, A245834, A245835, A216973.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Exp[# Exp[#]]&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

/* By definition C^C: */
{T(n,k)=local(A, C=matrix(n+1,n+1,r,c,binomial(r-1,c-1)), L=matrix(n+1,n+1,r,c,if(r==c+1,c))); A=sum(m=0,n,L^m*C^m/m!); A[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From e.g.f.: */
{T(n,k)=local(A=1);A=exp( x*y + x*exp(x +x*O(x^n)) );n!*polcoeff(polcoeff(A, n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From o.g.f. (Paul D. Hanna, Aug 03 2014): */
{T(n,k)=local(A=1);A=sum(k=0, n, x^k/(1 - x*(k+y) +x*O(x^n))^(k+1));polcoeff(polcoeff(A, n,x),k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From row polynomials (Paul D. Hanna, Aug 03 2014): */
{T(n,k)=local(R);R=sum(k=0,n,(k+y)^(n-k)*binomial(n,k));polcoeff(R,k,y)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* From formula for T(n,k) (Paul D. Hanna, Aug 03 2014): */
{T(n,k) = sum(j=0,n-k, binomial(n,j) * binomial(n-j,k) * j^(n-k-j))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

E.g.f.: exp( x*exp(x) + x*y ).
From Peter Bala, Sep 13 2012: (Start)
Exponential Riordan array [exp(x*exp(x)),x] belonging to the Appell group. Thus the e.g.f. for the k-th column of the triangle is x^k/k!*exp(x*exp(x)).
The inverse array, denote it by X, is a signed version of A215652. The infinite tower of matrix powers X^(X^(X^(...))) equals the inverse of Pascal's triangle. (End)
O.g.f.: Sum_{n>=0} x^n / (1 - x*(n+y))^(n+1). - Paul D. Hanna, Aug 03 2014
G.f. for row n: Sum_{k=0..n} binomial(n,k) * (k + y)^(n-k) for n>=0. - Paul D. Hanna, Aug 03 2014
T(n,k) = Sum_{j=0..n-k} C(n,j) * C(n-j,k) * j^(n-k-j) = A000248(n-k)*C(n,k). - Paul D. Hanna, Aug 03 2014
Infinitesimal generator is A216973. - Peter Bala, Feb 13 2017

A245835 E.g.f.: exp( x(2 + exp(3x)) ).

Original entry on oeis.org

1, 3, 15, 108, 945, 9558, 109917, 1412316, 19959777, 306805482, 5087064789, 90370321704, 1710170426097, 34308056537550, 726612812416269, 16188742781216892, 378244417385086785, 9242436410233527762, 235609985190361119525, 6252379688953421699760, 172380307421633200750161

Offset: 0

Paul D. Hanna, Aug 02 2014

nonn

			E.g.f.: E(x) = 1 + 3*x + 15*x^2/2! + 108*x^3/3! + 945*x^4/4! + 9558*x^5/5! +...
where E(x) = exp(2*x) * exp(x*exp(3*x)).
O.g.f.: A(x) = 1 + 3*x + 15*x^2 + 108*x^3 + 945*x^4 + 9558*x^5 + 109917*x^6 +...
where
A(x) = 1/(1-2*x) + x/(1-5*x)^2 + x^2/(1-8*x)^3 + x^3/(1-11*x)^4 + x^4/(1-14*x)^5 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A080108, A216689, A240165, A245834.

Table[Sum[Binomial[n,k] * (3*k+2)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2014 *)
With[{nn=20},CoefficientList[Series[Exp[x(2+Exp[3x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 06 2015 *)

{a(n)=local(A=1);A=exp( x*(2 + exp(3*x +x*O(x^n))) );n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - (3*k+2)*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

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

O.g.f.: Sum_{n>=0} x^n / (1 - (3*n+2)*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (3*k+2)^(n-k) for n>=0.
a(n) ~ exp((n+6*r^2)/(1+3*r)) * n! / (r^n*sqrt(2*Pi*(-6*r^2*(2+3*r) + n*(1+9*r+9*r^2)) / (1+3*r))), where r is the root of the equation r*(2 + (1+3*r)*exp(3*r)) = n. - Vaclav Kotesovec, Aug 03 2014
(a(n)/n!)^(1/n) ~ 3*exp(1/(2*LambertW(sqrt(3*n)/2))) / (2*LambertW(sqrt(3*n)/2)). - Vaclav Kotesovec, Aug 06 2014

A295552 a(n) = n! * [x^n] exp(xexp(nx)).

Original entry on oeis.org

1, 1, 5, 46, 689, 15476, 483157, 19719022, 1009495489, 63119450152, 4728073048901, 417482964953594, 42834647403146161, 5043607239173464924, 674409403861210214485, 101517071981284179924526, 17074451852556909059698433, 3187883879639402167714593488, 656838643288782957496595002117

Offset: 0

Ilya Gutkovskiy, Nov 23 2017

nonn

N. J. A. Sloane, Transforms

Cf. A000248, A216689, A292914.

Table[n! SeriesCoefficient[Exp[x Exp[n x]], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[Binomial[n, k] (n k)^(n - k), {k, 0, n}], {n, 1, 18}]]

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

A356819 Expansion of e.g.f. exp(-x * exp(2*x)).

Original entry on oeis.org

1, -1, -3, -1, 41, 239, 229, -8401, -87151, -324577, 3238541, 70271519, 601086265, 142860431, -81504662539, -1393683935281, -10777424809951, 63537986981183, 3552608426329117, 60283510555017023, 441644419610814281, -6191820436867600081

Offset: 0

Seiichi Manyama, Aug 29 2022

sign

Cf. A216689, A292952, A356812, A356820.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*exp(2*x))))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-2*k*x)^(k+1)))

a(n) = sum(k=0, n, (-1)^k*(2*k)^(n-k)*binomial(n, k));

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

a(n) = Sum_{k=0..n} (-1)^k * (2*k)^(n-k) * binomial(n,k).

A356827 Expansion of e.g.f. exp(x * exp(3*x)).

Original entry on oeis.org

1, 1, 7, 46, 361, 3436, 37729, 463366, 6280369, 93015352, 1491337441, 25684077706, 472217487625, 9221588527204, 190441412508481, 4143470377262806, 94663498086222049, 2264440394856702832, 56570146384760433217, 1472545685988162638722

Offset: 0

Seiichi Manyama, Aug 29 2022

nonn

Cf. A000248, A003725, A216689, A295552.

Cf. A277456, A336951, A351737, A355501, A356820.

A356827 := proc(n)
    add((3*k)^(n-k) * binomial(n,k),k=0..n) ;
end proc:
seq(A356827(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(3*x))))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))

a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} x^k / (1 - 3*k*x)^(k+1).

a(n) = Sum_{k=0..n} (3*k)^(n-k) * binomial(n,k).

cp's OEIS Frontend

A216688 Expansion of e.g.f. exp( x * exp(x^2) ).

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A216507 E.g.f. exp( x^2 * exp(x) ).

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A240165 E.g.f.: exp( x*(1 + exp(2*x)) ).

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A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

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A245834 E.g.f.: exp( x*(1 + exp(3*x)) ).

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A116071 Triangle T, read by rows, equal to Pascal's triangle to the matrix power of Pascal's triangle, so that T = C^C, where C(n,k) = binomial(n,k) and T(n,k) = A000248(n-k)*C(n,k).

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A245835 E.g.f.: exp( x*(2 + exp(3*x)) ).

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A240165 E.g.f.: exp( x(1 + exp(2x)) ).

A245834 E.g.f.: exp( x(1 + exp(3x)) ).

A245835 E.g.f.: exp( x(2 + exp(3x)) ).

A295552 a(n) = n! * [x^n] exp(xexp(nx)).