A108459 -id:A108459 - cp's OEIS frontend

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

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.

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).

A124421 Number of partitions of the set {1,2,...,n} having no blocks that contain only odd entries.

Original entry on oeis.org

1, 0, 1, 1, 5, 9, 52, 130, 855, 2707, 19921, 75771, 614866, 2717570, 24040451, 120652827, 1152972925, 6460552857, 66200911138, 408845736040, 4465023867757, 30083964854141, 348383154017581, 2539795748336375, 31052765897026352, 243282175672281360

Offset: 0

Emeric Deutsch, Oct 31 2006

nonn

Column 0 of A124420.

			a(4) = 5 because we have 1234, 134|2, 14|23, 12|34 and 123|4.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A000110, A124418, A124419, A124420, A124422, A124423.

Bisection gives A108459 (even part).

Q[0]:=1: for n from 1 to 27 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 27 do Q[n]:=Q[n] od: seq(subs({t=0,s=1,x=1},Q[n]),n=0..27);
# second Maple program:
a:= n-> add(Stirling2(floor(n/2), j)*j^ceil(n/2), j=0..floor(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

a[0] = 1; a[n_] := Sum[StirlingS2[Floor[n/2], j]*j^Ceiling[n/2], {j, 0, Floor[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2015, after Alois P. Heinz *)

a(n) = Q[n](0,1,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.

a(n) = Sum_{j=0..floor(n/2)} Stirling2(floor(n/2),j) * j^ceiling(n/2). - Alois P. Heinz, Oct 23 2013

A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(nx) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 2, 1, 15, 12, 2, 203, 206, 60, 5, 4140, 4949, 1947, 298, 15, 115975, 156972, 75595, 16160, 1535, 52, 4213597, 6301550, 3528368, 945360, 127915, 8307, 203, 190899322, 310279615, 195764198, 62079052, 10690645, 1001567, 47397, 877, 10480142147, 18293310174, 12735957930, 4614975428, 952279230, 114741060, 7901236, 285096, 4140, 682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147, 51724158235372, 101557600812015, 82635818516305, 36672690416280, 9831937482310, 1665456655065, 180791918475, 12443391060, 520878315, 12004575, 115975

Offset: 0

Paul D. Hanna, Jul 20 2019

			E.g.f.: A(x,y) = 1 + (2 + y)*x + (15 + 12*y + 2*y^2)*x^2/2! + (203 + 206*y + 60*y^2 + 5*y^3)*x^3/3! + (4140 + 4949*y + 1947*y^2 + 298*y^3 + 15*y^4)*x^4/4! + (115975 + 156972*y + 75595*y^2 + 16160*y^3 + 1535*y^4 + 52*y^5)*x^5/5! + (4213597 + 6301550*y + 3528368*y^2 + 945360*y^3 + 127915*y^4 + 8307*y^5 + 203*y^6)*x^6/6! + (190899322 + 310279615*y + 195764198*y^2 + 62079052*y^3 + 10690645*y^4 + 1001567*y^5 + 47397*y^6 + 877*y^7)*x^7/7! + (10480142147 + 18293310174*y + 12735957930*y^2 + 4614975428*y^3 + 952279230*y^4 + 114741060*y^5 + 7901236*y^6 + 285096*y^7 + 4140*y^8)*x^8/8! + (682076806159 + 1267153412532*y + 959061013824*y^2 + 387848415927*y^3 + 92381300277*y^4 + 13455280629*y^5 + 1200540180*y^6 + 63424134*y^7 + 1805067*y^8 + 21147*y^9)*x^9/9! + (51724158235372 + 101557600812015*y + 82635818516305*y^2 + 36672690416280*y^3 + 9831937482310*y^4 + 1665456655065*y^5 + 180791918475*y^6 + 12443391060*y^7 + 520878315*y^8 + 12004575*y^9 + 115975*y^10)*x^10/10! + ...
such that
A(x,y) = exp(-1-y) * (1 + (exp(x) + y) + (exp(2*x) + y)^2/2! + (exp(3*x) + y)^3/3! + (exp(4*x) + y)^4/4! + (exp(5*x) + y)^5/5! + (exp(6*x) + y)^6/6! + ...)
also
A(x,y) = exp(-1-y) * (exp(y) + exp(x)*exp(y*exp(x)) + exp(4*x)*exp(y*exp(2*x))/2! + exp(9*x)*exp(y*exp(3*x))/3! + exp(16*x)*exp(y*exp(4*x))/4! + exp(25*x)*exp(y*exp(5*x))/5! + exp(36*x)*exp(y*exp(6*x))/6! + ...).
This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(x,y) begins:
[1],
[2, 1],
[15, 12, 2],
[203, 206, 60, 5],
[4140, 4949, 1947, 298, 15],
[115975, 156972, 75595, 16160, 1535, 52],
[4213597, 6301550, 3528368, 945360, 127915, 8307, 203],
[190899322, 310279615, 195764198, 62079052, 10690645, 1001567, 47397, 877],
[10480142147, 18293310174, 12735957930, 4614975428, 952279230, 114741060, 7901236, 285096, 4140],
[682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147], ...
Main diagonal is A000110 (Bell numbers).
Leftmost column is A020557(n) = A000110(2*n), for n >= 0.
Row sums form A326433.

Paul D. Hanna, Table of n, a(n) for n = 0..495 (first 30 rows of this triangle).

Cf. A000110, A020557, A108459, A326433, A326434, A326435, A326436, A326437.

Cf. A326601 (central terms).

E.g.f.: exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!.
E.g.f.: exp(-1-y) * Sum_{n>=0} exp(n^2*x) * exp( y*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
T(n,n) = A000110(n) for n >= 0, where A000110 is the Bell numbers.
T(n,0) = A000110(2*n) for n >= 0, where A000110 is the Bell numbers.
Sum_{k=0..n} T(n,k) * (-1)^k = A108459(n) for n >= 0.
Sum_{k=0..n} T(n,k) = A326433(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 2^k = A326434(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 3^k = A326435(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 4^k = A326436(n) for n >= 0.

A051707 Number of factorizations of (n,n) into pairs (j,k).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 8, 3, 5, 1, 23, 1, 5, 5, 23, 1, 23, 1, 23, 5, 5, 1, 91, 3, 5, 8, 23, 1, 52, 1, 60, 5, 5, 5, 143, 1, 5, 5, 91, 1, 52, 1, 23, 23, 5, 1, 328, 3, 23, 5, 23, 1, 91, 5, 91, 5, 5, 1, 339, 1, 5, 23, 161, 5, 52, 1, 23, 5, 52, 1, 686, 1, 5, 23, 23, 5, 52, 1, 328, 23, 5, 1, 339, 5

Offset: 1

Yasutoshi Kohmoto

Pairs (j,k) must satisfy j>1, k>=1; (a,b)*(x,y)=(a*x,b*y); unit is (1,1).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

			(6,6)=(2,1)*(3,6)=(2,6)*(3,1)=(2,2)*(3,3)=(2,3)*(3,2), so a(6)=5.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A050354, A108461, A108455, A348161 (into at most two pairs).
a(A025487) = A108460.
a(p^k) = A108457(k).
a(A002110) = A108459.
Main diagonal of A108455.

Edited by Christian G. Bower, Jun 03 2005

A320082 Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k/k!.

Original entry on oeis.org

1, 1, 3, 5, -60, -186, 13832, -98862, -8631360, 352796880, 4245955032, -1185349047048, 48595690153920, 3201334718188320, -607575977909763840, 26489851912606455504, 4482546578798646251520, -958939334596403708474880, 50300999315063602037775360, 14223928928980522264922223360, -3933112779003946549567400925696

Offset: 0

Ilya Gutkovskiy, Oct 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A048994, A108459, A128943, A232549.

1,seq(n!*coeff(series(add(log(1+k*x)^k/k!, k=1..100), x=0, 21), x, n), n=1..20); # Paolo P. Lava, Jan 09 2019

nmax = 20; CoefficientList[Series[1 + Sum[Log[1 + k x]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^n, {k, n}], {n, 20}]]

a(n) = sum(k=0, n, stirling(n,k)*k^n); \\ Altug Alkan, Oct 05 2018

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

A326090 E.g.f.: Sum_{n>=0} (1 + exp(nx))^n x^n/n!.

Original entry on oeis.org

1, 2, 6, 35, 308, 3637, 55150, 1033027, 23260536, 617066297, 18968614874, 666664879663, 26496140541700, 1179815542970053, 58388906382906390, 3189604848766578563, 191168734534622234480, 12504288586619417431921, 888401197086798248554546, 68270033412187747029025111, 5652853046029263008213465916, 502601914954325406783531231677, 47834047958592244085651443711406

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 = x.

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 35*x^3/3! + 308*x^4/4! + 3637*x^5/5! + 55150*x^6/6! + 1033027*x^7/7! + 23260536*x^8/8! + 617066297*x^9/9! + 18968614874*x^10/10! + ...
such that
A(x) = 1 + (1 + exp(x))*x + (1 + exp(2*x))^2*x^2/2! + (1 + exp(3*x))^3*x^3/3! + (1 + exp(4*x))^4*x^4/4! + (1 + exp(5*x))^5*x^5/5! + (1 + exp(6*x))^6*x^6/6! + ...
also
A(x) = exp(x) + exp(x + exp(x)*x)*x + exp(4*x + exp(2*x)*x)*x^2/2! + exp(9*x + exp(3*x)*x)*x^3/3! + exp(16*x + exp(4*x)*x)*x^4/4! + exp(25*x + exp(5*x)*x)*x^5/5! + exp(36*x + exp(6*x)*x)*x^6/6! + ...
RELATED SERIES.
Below we illustrate the following identity at specific values of x:
Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!  =  Sum_{n>=0} exp(n^2*x) * exp( exp(n*x)*x ) * x^n/n!.
(1) At x = -1, the following sums are equal
S1 = Sum_{n>=0} (1 + exp(-n))^n * (-1)^n/n!,
S1 = Sum_{n>=0} exp(-n^2) * exp( -exp(-n) ) * (-1)^n/n!,
where S1 = 0.12121214669421724219987424741512642137552627624687959194...
(2) At x = -log(2), the following sums are equal
S2 = Sum_{n>=0} (1 + 1/2^n)^n * log(1/2)^n/n!,
S2 = Sum_{n>=0} 2^(-n^2) * 2^(-1/2^n) * log(1/2)^n/n!,
where S2 = 0.26746154600304489791062659014323146833150028333177021587...

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

Cf. A108459, A326091, A326261, A326009.

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

/* E.g.f.: Sum_{n>=0} exp( n^2*x + exp(n*x)*x ) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m^2*x + exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

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

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

A192935 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n / n!.

Original entry on oeis.org

1, 1, 4, 39, 592, 12965, 378276, 14062363, 643946920, 35426253465, 2295988778440, 172565368741931, 14847924324645996, 1446814927797156541, 158201328106874927980, 19258822568210913998955, 2592339296719295037808336, 383513887126740027040942577

Offset: 0

Paul D. Hanna, Jul 13 2011

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = (1+x) and p = -1, r = 1. - Paul D. Hanna, Jun 21 2019

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 39*x^3/3! + 592*x^4/4! + 12965*x^5/5! + 378276*x^6/6! + 14062363*x^7/7! + 643946920*x^8/8! + 35426253465*x^9/9! + 2295988778440*x^10/10! +...
such that
A(x) = 1 + ((1+x) - 1) + ((1+x)^2 - 1)^2/2! + ((1+x)^3 - 1)^3/3! + ((1+x)^4 - 1)^4/4! + ((1+x)^5 - 1)^5/5! + ((1+x)^6 - 1)^6/6! + ((1+x)^7 - 1)^7/7! + ...
also
A(x) = 1 + (1+x)*exp(-(1+x)) + (1+x)^4*exp(-(1+x)^2)/2! + (1+x)^9*exp(-(1+x)^3)/3! + (1+x)^16*exp(-(1+x)^4)/4! + (1+x)^25*exp(-(1+x)^5)/5! + (1+x)^36*exp(-(1+x)^6)/6! + (1+x)^49*exp(-(1+x)^7)/7! + ...
RELATED SERIES.
Expansion of ((1+x)^n-1)^n suggests that the e.g.f. is related to LambertW(x):
((1+x)^2-1)^2 = 4*x^2 + 4*x^3 + x^4;
((1+x)^3-1)^3 = 27*x^3 + 81*x^4 + 108*x^5 + 81*x^6 + 36*x^7 + 9*x^8 + x^9;
((1+x)^4-1)^4 = 256*x^4 + 1536*x^5 + 4480*x^6 + 8320*x^7 + 10896*x^8 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..280

Cf. A192985, A108459, A122400.

m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[((1+x)^n -1)^n/Factorial(n): n in [0..m+2]]) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 06 2019

With[{m = 20}, CoefficientList[Series[Sum[If[n==0, 1, ((1+x)^n -1)^n/n!], {n,0,m+2}], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 06 2019 *)

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

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0, n, Stirling1(n, k)*k!*polcoeff(sum(m=0,k,(exp(m*x+x*O(x^n))-1)^m/m!),k))}
for(n=0, 30, print1(a(n)*n!, ", "))

m = 20; T = taylor(sum(((1+x)^k-1)^k/factorial(k) for k in range(m+2)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 06 2019

E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(-(1+x)^n) / n!. - Paul D. Hanna, Jun 21 2019

a(n) = (1/n!)* Sum_{k=0..n} Stirling1(n,k)*A108459(k), where the e.g.f. of A108459 = Sum_{n>=0} (exp(n*x)-1)^n/n! (see Vladeta Jovovic's formula in A122400).

A326091 E.g.f.: Sum_{n>=0} (2 + exp(nx))^n x^n/n!.

Original entry on oeis.org

1, 3, 11, 66, 601, 7418, 116505, 2248522, 52025473, 1414524690, 44471074249, 1595792690594, 64659403375137, 2931455146804330, 147550017664392457, 8189594420467104042, 498288959815836863233, 33061714451161940667554, 2381086262720126177230473, 185362512554618232339122578, 15539467373234774634135507361, 1398111233425766921500901239098, 134584560980879138160145116701257

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 = 2, r = x.

			E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 66*x^3/3! + 601*x^4/4! + 7418*x^5/5! + 116505*x^6/6! + 2248522*x^7/7! + 52025473*x^8/8! + 1414524690*x^9/9! + 44471074249*x^10/10! + ...
such that
A(x) = 1 + (2 + exp(x))*x + (2 + exp(2*x))^2*x^2/2! + (2 + exp(3*x))^3*x^3/3! + (2 + exp(4*x))^4*x^4/4! + (2 + exp(5*x))^5*x^5/5! + (2 + exp(6*x))^6*x^6/6! + ...
also
A(x) = exp(2*x) + exp(x + 2*exp(x)*x)*x + exp(4*x + 2*exp(2*x)*x)*x^2/2! + exp(9*x + 2*exp(3*x)*x)*x^3/3! + exp(16*x + 2*exp(4*x)*x)*x^4/4! + exp(25*x + 2*exp(5*x)*x)*x^5/5! + exp(36*x + 2*exp(6*x)*x)*x^6/6! + ...

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

Cf. A108459, A326090, A326261.

/* E.g.f.: Sum_{n>=0} (2 + exp(n*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (2 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* E.g.f.: Sum_{n>=0} exp( n^2*x + 2*exp(n*x)*x ) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m^2*x + 2*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

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

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

A326261 E.g.f.: Sum_{n>=0} (3 + exp(nx))^n x^n/n!.

Original entry on oeis.org

1, 4, 18, 115, 1076, 13749, 223342, 4437115, 105308472, 2930229721, 94110395546, 3444510650343, 142161931150564, 6557368148307253, 335460464343013494, 18907437932151629899, 1167279375125285092592, 78529603970775837111729, 5730854443905658384812466, 451803953552256670477653679, 38337003901469883140928003036, 3489532046271886600931347767373

Offset: 0

Paul D. Hanna, Jun 29 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 = 3, r = x.

			E.g.f.: A(x) = 1 + 4*x + 18*x^2/2! + 115*x^3/3! + 1076*x^4/4! + 13749*x^5/5! + 223342*x^6/6! + 4437115*x^7/7! + 105308472*x^8/8! + 2930229721*x^9/9! + 94110395546*x^10/10! + ...
such that
A(x) = 1 + (3 + exp(x))*x + (3 + exp(2*x))^2*x^2/2! + (3 + exp(3*x))^3*x^3/3! + (3 + exp(4*x))^4*x^4/4! + (3 + exp(5*x))^5*x^5/5! + (3 + exp(6*x))^6*x^6/6! + ...
also
A(x) = exp(3*x) + exp(x + 3*exp(x)*x)*x + exp(4*x + 3*exp(2*x)*x)*x^2/2! + exp(9*x + 3*exp(3*x)*x)*x^3/3! + exp(16*x + 3*exp(4*x)*x)*x^4/4! + exp(25*x + 3*exp(5*x)*x)*x^5/5! + exp(36*x + 3*exp(6*x)*x)*x^6/6! + ...

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

Cf. A108459, A326090, A326091.

/* E.g.f.: Sum_{n>=0} (3 + exp(n*x))^n * x^n/n! */
{a(n) = my(A = sum(m=0, n, (3 + exp(m*x +x*O(x^n)))^m * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* E.g.f.: Sum_{n>=0} exp( n^2*x + 3*exp(n*x)*x ) * x^n/n! */
{a(n) = my(A = sum(m=0, n, exp(m^2*x + 3*exp(m*x +x*O(x^n))*x ) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

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

E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x)*x ) * x^n/n!.

cp's OEIS Frontend

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

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

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A124421 Number of partitions of the set {1,2,...,n} having no blocks that contain only odd entries.

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A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

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A051707 Number of factorizations of (n,n) into pairs (j,k).

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A320082 Expansion of e.g.f. Sum_{k>=0} log(1 + k*x)^k/k!.

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A326090 E.g.f.: Sum_{n>=0} (1 + exp(n*x))^n * x^n/n!.

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

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

A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(nx) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

A326090 E.g.f.: Sum_{n>=0} (1 + exp(nx))^n x^n/n!.

A326091 E.g.f.: Sum_{n>=0} (2 + exp(nx))^n x^n/n!.

A326261 E.g.f.: Sum_{n>=0} (3 + exp(nx))^n x^n/n!.