A216688 -id:A216688 - cp's OEIS frontend

A354553 Expansion of e.g.f. exp( x * exp(x^3) ).

1, 1, 1, 1, 25, 121, 361, 3361, 42001, 275185, 1819441, 30777121, 371238121, 3057311401, 44263763545, 801096528961, 9710981323681, 125367419194081, 2643123767954401, 45840730383002305, 646414025466298681, 13258301279836276441

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..475

Cf. A000248, A216688, A354554.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^3)))))

a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(k!*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(k! * (n - 3*k)!).

A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(n - 2k)!.

Original entry on oeis.org

1, 1, 1, 7, 25, 181, 1561, 12811, 188497, 2071945, 38889361, 620762671, 12917838121, 291278938237, 6667342764265, 194869722610291, 5137978752994081, 177509783765281681, 5610285632192738977, 215195998789004395735, 8228064506323330305721

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A354436, A356629, A356630.

Cf. A216688, A356632, A358064.

a[n_] := n! * Sum[(n - 2*k)^k/(n - 2*k)!, {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 19 2022 *)

a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(n-2*k)!);

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

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

a(n) ~ sqrt(Pi) * exp((n-1)/(2*LambertW(exp(1/3)*(n-1)/3)) - 3*n/2) * n^((3*n + 1)/2) / (sqrt(1 + LambertW(exp(1/3)*(n - 1)/3)) * 3^((n+1)/2) * LambertW(exp(1/3)*(n-1)/3)^(n/2)). - Vaclav Kotesovec, Nov 01 2022

A009229 Expansion of e.g.f. exp(sinh(x)*cosh(x)).

Original entry on oeis.org

1, 1, 1, 5, 17, 57, 337, 1661, 9697, 68529, 451809, 3570613, 29324913, 247557481, 2348275633, 22399292077, 229227939521, 2476740370529, 27378158737089, 323000223133285, 3912000089144273, 49350756905258905, 649905519911431953, 8776449485870211549

Offset: 0

R. H. Hardin

Alois P. Heinz, Table of n, a(n) for n = 0..528 (first 126 terms from Vincenzo Librandi)
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
      a(n-j)*binomial(n-1, j-1)*2^(j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 22 2021

With[{nn=30},CoefficientList[Series[Exp[Sinh[x]Cosh[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 23 2011 *)

a(n):=sum(((sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k))*2^(n-2*k))/k!,k,1,n); /* Vladimir Kruchinin, May 29 2011 */

x='x+O('x^66); /* that many terms */
Vec(serlaplace(exp(sinh(x)*cosh(x)))) /* show terms */ /* Joerg Arndt, May 29 2011 */

a(n) = sum(k=1..n, ((sum(i=0..k, (-1)^i*(k-2*i)^n*binomial(k,i)))*2^(n-2*k))/k!), n>0, a(0)=1. - Vladimir Kruchinin, May 29 2011
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp(cosh(r)*sinh(r)-n) * n^(n+1/2) / (sqrt(cosh(2*r) + 2*r*sinh(2*r)) * r^(n+1/2)), where r is the root of the equation r*(cosh(r)^2 + sinh(r)^2) = n.
(a(n)/n!)^(1/n) ~ 2*(exp(1/LambertW(4*n))/LambertW(4*n)).
(End)
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 4^k * a(n-2*k-1). - Ilya Gutkovskiy, Feb 24 2022

Extended and signs tested by Olivier Gérard, Mar 15 1997

Corrected definition, Joerg Arndt, May 29 2011

A354550 Expansion of e.g.f. exp( x * exp(x^2/2) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 46, 241, 1156, 6889, 44668, 300241, 2328976, 18390901, 159273544, 1461200833, 13995753136, 144068872081, 1531949061136, 17259159775969, 202543867724608, 2474236899786781, 31633380519660256, 417760492214548561, 5751414293905728064

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..518

Cf. A000248, A354551, A354552.

Cf. A216688.

With[{nn=30},CoefficientList[Series[Exp[x Exp[x^2/2]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 03 2024 *)

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

a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(2^k*k!*(n-2*k)!));

a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(2^k * k! * (n - 2*k)!).

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)

A009153 Expansion of e.g.f. cosh(sinh(x)*exp(x)).

Original entry on oeis.org

1, 0, 1, 6, 29, 140, 757, 4858, 36409, 302520, 2681769, 25018510, 245905365, 2559272196, 28264854685, 330408571202, 4065526003313, 52349977261040, 702393407898705, 9795673312888214, 141820637175889805

Offset: 0

R. H. Hardin

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A009599, A065143.

CoefficientList[Series[Cosh[Sinh[x]*E^x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 06 2014 *)
Table[Sum[StirlingS2[n, k]*(1+(-1)^k)/2*2^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 after Vladeta Jovovic *)
Table[(BellB[n, 1/2] + BellB[n, -1/2]) 2^(n-1), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

a(n) = sum(k=0, n, stirling(n, k, 2)*(1+(-1)^k)/2*2^(n-k)); \\ Michel Marcus, Nov 02 2015

a(n) = Sum_{k=0..n} Stirling2(n, k)*(1+(-1)^k)/2*2^(n-k). - Vladeta Jovovic, Sep 26 2003
G.f.: 1 + Sum_{k>=0} x^(2*k+2)/Product_{i=0..2*k+2} (1-2*i*x). - Sergei N. Gladkovskii, Jan 06 2013
G.f.: 1 + x^2/( G(0)-x^2 ) where G(k) = x^2 + (4*x*k+2*x-1)*(4*x*k+4*x-1) - x^2*(4*x*k+2*x-1)*(4*x*k+4*x-1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 06 2013
a(n) ~ cosh(exp(r)*sinh(r)) * n^(n+1/2) / (r^(n+1/2) * exp(n+r) * sqrt(exp(2*r) * r * sech(exp(r)*sinh(r))^2 + (1+2*r) * tanh(exp(r)*sinh(r)))), where r is the root of the equation r*exp(r)*(cosh(r) + sinh(r))*tanh(exp(r)*sinh(r)) = n. - Vaclav Kotesovec, Aug 06 2014

Extended and signs tested by Olivier Gérard, Mar 15 1997

A354554 Expansion of e.g.f. exp( x * exp(x^4) ).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 196561, 3659041, 29993041, 159762241, 1686639241, 60298558321, 987112886761, 9315623640961, 76611297104161, 2454331471018561, 69805324167893281, 1086439146068753281, 11530308934656915481

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..467

Cf. A000248, A216688, A354553.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^4)))))

a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(k!*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4*k)^k/(k! * (n - 4*k)!).

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

A357948 Expansion of e.g.f. exp( x * exp(-x^2) ).

Original entry on oeis.org

1, 1, 1, -5, -23, 1, 601, 2731, -13775, -219743, -313199, 15383611, 125451481, -811558175, -20767068503, -37852036949, 2898343066081, 28990920216001, -313289894357855, -8634009894555653, -3214642669500599, 2108734127922999361, 20183394611962437241

Offset: 0

Seiichi Manyama, Oct 29 2022

Cf. A003725, A358063.

Cf. A216688.

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

a(n) = n!*sum(k=0, n\2, (-n+2*k)^k/(k!*(n-2*k)!));

a(n) = n! * Sum_{k=0..floor(n/2)} (-n + 2*k)^k/(k! * (n - 2*k)!).

A362604 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^2))).

Original entry on oeis.org

1, 1, 4, 33, 352, 4805, 80256, 1582693, 36001792, 927974601, 26729943040, 850921057481, 29666297020416, 1124166449205709, 46005243970846720, 2022121401647311245, 95008417631810093056, 4751844218849365365137, 252063937292253895065600

Offset: 0

Seiichi Manyama, Apr 30 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A072034, A362699.

Cf. A216688, A356834, A362700, A362702.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x*exp(x^2)))))

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^(n-k) / (k! * (n-2*k)!).

cp's OEIS Frontend

A354553 Expansion of e.g.f. exp( x * exp(x^3) ).

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A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(n - 2*k)!.

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A009229 Expansion of e.g.f. exp(sinh(x)*cosh(x)).

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A354550 Expansion of e.g.f. exp( x * exp(x^2/2) ).

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

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A009153 Expansion of e.g.f. cosh(sinh(x)*exp(x)).

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A354554 Expansion of e.g.f. exp( x * exp(x^4) ).

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

A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(n - 2k)!.

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

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