A240165 -id:A240165 - cp's OEIS frontend

A216689 Expansion of e.g.f. exp( x * exp(x)^2 ).

1, 1, 5, 25, 153, 1121, 9373, 87417, 898033, 10052353, 121492341, 1573957529, 21729801481, 318121178337, 4917743697805, 79981695655801, 1364227940101857, 24335561350365953, 452874096174214117, 8772713803852981785, 176541611843378273401, 3684142819311127955041, 79596388271096140589949

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))), A216688 (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))).
Cf. A240165 (e.g.f. exp(x*(1+exp(x)^2))).

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 */

/* From o.g.f.: */
{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - 2*k*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

/* From binomial sum: */
{a(n)=sum(k=0,n, binomial(n,k)*(2*k)^(n-k))}
for(n=0,30,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

O.g.f.: Sum_{n>=0} x^n / (1 - 2*n*x)^(n+1). - Paul D. Hanna, Aug 02 2014
a(n) = Sum_{k=0..n} binomial(n,k) * (2*k)^(n-k) for n>=0. - Paul D. Hanna, Aug 02 2014
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(2*n*r/(1+2*r)) * r^n * sqrt((1+6*r+4*r^2)/(1+2*r))), where r is the root of the equation r*(1+2*r)*exp(2*r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
(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)

A356815 Expansion of e.g.f. exp(-x * (exp(2*x) + 1)).

Original entry on oeis.org

1, -2, 0, 4, 32, 48, -608, -6400, -24064, 163072, 3567104, 28394496, 6535168, -3250745344, -50725740544, -344530853888, 2476610551808, 110057610608640, 1655672654135296, 9616664975114240, -195178079811272704, -6998474114188967936, -110894925369151848448

Offset: 0

Seiichi Manyama, Aug 29 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..499

Cf. A356816, A356818.

Cf. A240165, A351736, A356812.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(2*x)+1))))

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

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

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

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

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

A356811 a(n) = Sum_{k=0..n} (kn+1)^(n-k) binomial(n,k).

Original entry on oeis.org

1, 2, 8, 71, 1040, 22457, 676000, 26861977, 1347932416, 82873789793, 6114540967424, 532596023373713, 53990083205042176, 6289985311473281329, 833180470332123750400, 124356049859476364116193, 20754548375601491155681280, 3847574240184742568296430273

Offset: 0

Seiichi Manyama, Aug 29 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..274

Cf. A080108, A240165, A245834.

Cf. A356806, A356814, A356817.

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

a(n) = n! * [x^n] exp( x * (exp(n * x) + 1) ).

a(n) = [x^n] Sum_{k>=0} x^k / (1 - (n*k+1)*x)^(k+1).

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

cp's OEIS Frontend

A216689 Expansion of e.g.f. exp( x * exp(x)^2 ).

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

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A356815 Expansion of e.g.f. exp(-x * (exp(2*x) + 1)).

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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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A356811 a(n) = Sum_{k=0..n} (k*n+1)^(n-k) * binomial(n,k).

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

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A367874 Expansion of e.g.f. exp(x * (2 + exp(x))).

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

A356811 a(n) = Sum_{k=0..n} (kn+1)^(n-k) binomial(n,k).

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