A217900 -id:A217900 - cp's OEIS frontend

A220955 O.g.f.: Sum_{n>=0} (2n+1)^(2n+1) * exp(-(2n+1)^2x) * x^n / n!.

1, 26, 1320, 99288, 9901920, 1230768704, 183260197120, 31800433551744, 6301891570411008, 1404224096732154880, 347532097449969496064, 94584986134590717358080, 28076463606243146379018240, 9027122730610037995425792000, 3125219575155651450096795648000

Offset: 0

Paul D. Hanna, Feb 27 2013

nonn

From Vaclav Kotesovec, May 13 2014: (Start)
Generally, for p>1, a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (p*k+1)^(p*n+1) = Sum_{k=0..(p-1)*n+1} p^(n+k) * binomial(p*n+1,n+k) * stirling2(n+k,n).
a(n) ~ n^(n*p-n+1/2) * p^(2*p*n+2+1/p) / (sqrt(2*Pi*(1-r)) * exp((p-1)*n) * r^(n+1/p) * (p-r)^(n*p-n+1)), where r = -LambertW(-p*exp(-p)).
(End)

			O.g.f.: A(x) = 1 + 26*x + 1320*x^2 + 99288*x^3 + 9901920*x^4 +...
where A(x) = exp(-x) + 3^3*exp(-3^2*x)*x + 5^5*exp(-5^2*x)*x^2/2! + 7^7*exp(-7^2*x)*x^3/3! + 9^9*exp(-9^2*x)*x^4/4! + 11^11*exp(-11^2*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..290

Cf. A222525, A221214, A213193, A217900, A007820, A242449.

Table[1/n! * Sum[(-1)^(n-k)*Binomial[n,k] * (2*k+1)^(2*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[2*n+1,n+k]*2^(n+k)*StirlingS2[n+k,n],{k,0,n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*exp(-(2*k+1)^2*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (2*k+1)^(2*k+1)*x^k/(1+(2*k+1)^2*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(2*k+1)^(2*n+1))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (2*k+1)^(2*k+1) * x^k / (1 + (2*k+1)^2*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (2*k+1)^(2*n+1).
a(n) = Sum_{k=0..n+1} 2^(n+k) * binomial(2*n+1,n+k) * stirling2(n+k,n). - Vaclav Kotesovec, May 13 2014
a(n) ~ n^(n+1/2) * 2^(4*n+5/2) / (sqrt(2*Pi*(1-r)) * exp(n) * r^(n+1/2) * (2-r)^(n+1)), where r = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775 = -r) . - Vaclav Kotesovec, May 13 2014

A221214 O.g.f.: Sum_{n>=0} (3n+1)^(3n+1) * exp(-(3n+1)^3x) * x^n / n!.

Original entry on oeis.org

1, 255, 395388, 1525953330, 10977340509135, 126827739333023274, 2148335345336441463090, 50163717301669569182864400, 1544377393328765493716910877185, 60615459491155396034172113103266025, 2954227738557038665136475801709196246304

Offset: 0

Paul D. Hanna, Feb 27 2013

nonn

			O.g.f.: A(x) = 1 + 255*x + 395388*x^2 + 1525953330*x^3 + 10977340509135*x^4 +...
where A(x) = exp(-x) + 4^4*x*exp(-4^3*x) + 7^7*exp(-7^3*x)*x^2/2! + 10^10*exp(-10^3*x)*x^3/3! + 13^13*exp(-13^3*x)*x^4/4! + 16^16*exp(-16^3*x)*x^5/5! +... is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..159

Cf. A220955, A213193, A217900, A007820, A242449.

Table[1/n!*Sum[(-1)^(n-k)*Binomial[n,k]*(3*k+1)^(3*n+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)
Table[Sum[Binomial[3*n+1,n+k]*3^(n+k)*StirlingS2[n+k,n],{k,0,2*n+1}],{n,0,20}] (* Vaclav Kotesovec, May 13 2014 *)

{a(n)=polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*exp(-(3*k+1)^3*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=(1/n!)*polcoeff(sum(k=0, n, (3*k+1)^(3*k+1)*x^k/(1+(3*k+1)^3*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(3*k+1)^(3*n+1))}
for(n=0, 20, print1(a(n), ", "))

a(n) = 1/n! * [x^n] Sum_{k>=0} (3*k+1)^(3*k+1) * x^k / (1 + (3*k+1)^3*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (3*k+1)^(3*n+1).
a(n) ~ n^(2*n+1/2) * 3^(6*n+7/3) / (sqrt(2*Pi*(1-r)) * exp(2*n) * r^(n+1/3) * (3-r)^(2*n+1)), where r = -LambertW(-3*exp(-3)) = 0.1785606278779211... (see A226750 = -r) . - Vaclav Kotesovec, May 13 2014

A222526 O.g.f.: Sum_{n>=0} (n^6)^n * exp(-n^6x) x^n / n!.

Original entry on oeis.org

1, 1, 2047, 64439010, 11681056634501, 7713000216608565075, 14204422416132896951197888, 61232072982330045410678351728440, 545827051514425992551826008968173372261, 9173647538352903119028122246836507680995590680

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

			O.g.f.: A(x) = 1 + x + 2047*x^2 + 64439010*x^3 + 11681056634501*x^4 +...+ Stirling2(6*n, n)*x^n +...
where
A(x) = 1 + 1^6*x*exp(-1^6*x) + 2^12*exp(-2^6*x)*x^2/2! + 3^18*exp(-3^6*x)*x^3/3! + 4^24*exp(-4^6*x)*x^4/4! + 5^30*exp(-5^6*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..96

Cf. A007820, A217913, A217914, A217915, A222527, A222528, A222529, A222530, A217900.

Table[StirlingS2[6*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^6)^k*exp(-k^6*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^6)^k*x^k/(1+k^6*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(5*n))), 5*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(6*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(6*n, n).
a(n) = [x^(6*n)] (6*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(5*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^6)^k*x^k / (1 + k^6*x)^(k+1).
a(n) ~ n^(5*n) * 6^(6*n) / (sqrt(2*Pi*(1-c)*n) * exp(5*n) * (6-c)^(5*n) * c^n), where c = -LambertW(-6*exp(-6)). - Vaclav Kotesovec, May 11 2014

A222527 O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7x) x^n / n!.

Original entry on oeis.org

1, 1, 8191, 1742343625, 2998587019946701, 24204004899040755811870, 666480349285726891499539272955, 50789872166903636182659702516635946082, 9237419992097529135737293866043969707761346313, 3590622358224471993651445012122431990834934483552661750

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

			O.g.f.: A(x) = 1 + x + 8191*x^2 + 1742343625*x^3 + 2998587019946701*x^4 +...+ Stirling2(7*n, n)*x^n +...
where
A(x) = 1 + 1^7*x*exp(-1^7*x) + 2^14*exp(-2^7*x)*x^2/2! + 3^21*exp(-3^7*x)*x^3/3! + 4^28*exp(-4^7*x)*x^4/4! + 5^35*exp(-5^7*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..83

Cf. A007820, A217913, A217914, A217915, A222526, A222528, A222529, A222530, A217900.

Table[StirlingS2[7*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^7)^k*exp(-k^7*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^7)^k*x^k/(1+k^7*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(6*n))), 6*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(7*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(7*n, n).
a(n) = [x^(7*n)] (7*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(6*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^7)^k*x^k / (1 + k^7*x)^(k+1).
a(n) ~ n^(6*n) * 7^(7*n) / (sqrt(2*Pi*(1-c)*n) * exp(6*n) * (7-c)^(6*n) * c^n), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 11 2014

A222528 O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8x) x^n / n!.

Original entry on oeis.org

1, 1, 32767, 47063200806, 768305500780164501, 75740854251732106906082250, 31154086963475828638359480518580526, 41929298560838945526242744414099901692285884, 155440114706926165785630654089245708839702615196926765, 1396002062838446082394548660243302585983358463911636390911298400

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

			O.g.f.: A(x) = 1 + x + 32767*x^2 + 47063200806*x^3 + 768305500780164501*x^4 +...+ Stirling2(8*n, n)*x^n +...
where
A(x) = 1 + 1^8*x*exp(-1^8*x) + 2^16*exp(-2^8*x)*x^2/2! + 3^24*exp(-3^8*x)*x^3/3! + 4^32*exp(-4^8*x)*x^4/4! + 5^40*exp(-5^8*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222529, A222530, A217900.

Table[StirlingS2[8*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^8)^k*exp(-k^8*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^8)^k*x^k/(1+k^8*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(7*n))), 7*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(8*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(8*n, n).
a(n) = [x^(8*n)] (8*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(7*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^8)^k*x^k / (1 + k^8*x)^(k+1).
a(n) ~ n^(7*n) * 8^(8*n) / (sqrt(2*Pi*(1-c)*n) * exp(7*n) * (8-c)^(7*n) * c^n), where c = -LambertW(-8*exp(-8)). - Vaclav Kotesovec, May 11 2014

A222529 O.g.f.: Sum_{n>=0} (n^9)^n * exp(-n^9x) x^n / n!.

Original entry on oeis.org

1, 1, 131071, 1270865805301, 196740254364198919901, 236795997997922560392792426501, 1454443713270449746545892977574122129433, 34559048315358253352594346952765431711799794270765, 2610516895723221966171633379256064857587637240616032299710417

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

			O.g.f.: A(x) = 1 + x + 131071*x^2 + 1270865805301*x^3 + 196740254364198919901*x^4 +...+ Stirling2(9*n, n)*x^n +...
where
A(x) = 1 + 1^9*x*exp(-1^9*x) + 2^18*exp(-2^9*x)*x^2/2! + 3^27*exp(-3^9*x)*x^3/3! + 4^36*exp(-4^9*x)*x^4/4! + 5^45*exp(-5^9*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..66

Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222528, A222530, A217900.

Table[StirlingS2[9*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^9)^k*exp(-k^9*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^9)^k*x^k/(1+k^9*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(8*n))), 8*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(9*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(9*n, n).
a(n) = [x^(9*n)] (9*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(8*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^9)^k*x^k / (1 + k^9*x)^(k+1).
a(n) ~ n^(8*n) * 9^(9*n) / (sqrt(2*Pi*(1-c)*n) * exp(8*n) * (9-c)^(8*n) * c^n), where c = -LambertW(-9*exp(-9)). - Vaclav Kotesovec, May 11 2014

A222530 O.g.f.: Sum_{n>=1} (n^10)^n * exp(-n^10x) x^n / n!.

Original entry on oeis.org

1, 1, 524287, 34314651811530, 50369882873307917364901, 740095864368253016271188139587625, 67872880319721869662486234870635119906757244, 28468832412072117193931250482560479429446507352468258480, 43812568949824405485262661429905291482204531455805230631187460302069

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

Generally, for p>=2 is StirlingS2(p*n,n) asymptotic to n^((p-1)*n) * p^(p*n) / (sqrt(2*Pi*(1-c)*n) * exp((p-1)*n) * (p-c)^((p-1)*n) * c^n), where c = -LambertW(-p*exp(-p)). - Vaclav Kotesovec, May 11 2014

			O.g.f.: A(x) = 1 + x + 524287*x^2 + 34314651811530*x^3 + 50369882873307917364901*x^4 +...+ Stirling2(10*n, n)*x^n +...
where
A(x) = 1 + 1^10*x*exp(-1^10*x) + 2^20*exp(-2^10*x)*x^2/2! + 3^30*exp(-3^10*x)*x^3/3! + 4^40*exp(-4^10*x)*x^4/4! + 5^50*exp(-5^10*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..60

Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222528, A222529, A217900.

Table[StirlingS2[10*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^10)^k*exp(-k^10*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^1)^k*x^k/(1+k^10*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(9*n))), 9*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(10*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(10*n, n).
a(n) = [x^(10*n)] (10*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(9*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^10)^k*x^k / (1 + k^10*x)^(k+1).
a(n) ~ n^(9*n) * 10^(10*n) / (sqrt(2*Pi*(1-c)*n) * exp(9*n) * (10-c)^(9*n) * c^n), where c = -LambertW(-10*exp(-10)). - Vaclav Kotesovec, May 11 2014

A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2x)) (1 + n^2*x)^n / n!.

Original entry on oeis.org

1, 3, 31, 520, 11991, 350889, 12428746, 516450792, 24619176153, 1323971052261, 79280864647205, 5231080689880500, 377062508515478306, 29479066783583059530, 2484534527715953700780, 224559818606249783480400, 21666961097367611148157815, 2222844864226101120054773295

Offset: 0

Paul D. Hanna, Jul 12 2014

nonn

Compare the g.f. to:
(1) Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).
(2) Sum_{n>=1} exp(-n^2*x) * n^(2*n) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n (A007820).

			G.f.: A(x) = 1 + 3*x + 31*x^2 + 520*x^3 + 11991*x^4 + 350889*x^5 +...
where
A(x) = exp(-1) + exp(-(1+x))*(1+x) + exp(-(1+2^2*x))*(1+2^2*x)^2/2!
+ exp(-(1+3^2*x))*(1+3^2*x)^3/3! + exp(-(1+4^2*x))*(1+4^2*x)^4/4!
+ exp(-(1+5^2*x))*(1+5^2*x)^5/5! + exp(-(1+6^2*x))*(1+6^2*x)^6/6!
+ exp(-(1+7^2*x))*(1+7^2*x)^7/7! + exp(-(1+8^2*x))*(1+8^2*x)^8/8! +...
simplifies to a power series in x with integer coefficients.

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (first 100 terms from Paul D. Hanna)

Cf. A049020, A245110.

Cf. A187655, A217899, A217900.

Table[SeriesCoefficient[Sum[E^(-(1+k^2*x))*(1+k^2*x)^k/k!,{k,0,Infinity}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 12 2014 *)

/* Must first set suitable precision */ \p300
{a(n)=local(A=1+x); A=suminf(k=0, exp(-(1+k^2*x)+x*O(x^n))*(1+k^2*x)^k/k!); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ c * d^n * (n-1)!, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168..., and c = 10.427337127699040838035... . - Vaclav Kotesovec, Jul 12 2014

a(n) = A049020(2n,n). - Alois P. Heinz, Aug 23 2017

A218673 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(2n)/n! exp(-nxA(n*x)^2).

Original entry on oeis.org

1, 1, 3, 20, 209, 3173, 67292, 1970761, 79764057, 4490097388, 354111363537, 39360693851404, 6193012446752244, 1383433132321835172, 439684769985895688173, 199116777197880585373014, 128631139424158036273736167, 118640007280899188486618513612

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

			O.g.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 209*x^4 + 3173*x^5 + 67292*x^6 +...
where
A(x) = 1 + x*A(x)^2*exp(-x*A(x)^2) + 2^2*x^2*A(2*x)^4/2!*exp(-2*x*A(2*x)^2) + 3^3*x^3*A(3*x)^6/3!*exp(-3*x*A(3*x)^2) + 4^4*x^4*A(4*x)^8/4!*exp(-4*x*A(4*x)^2) + 5^5*x^5*A(5*x)^10/5!*exp(-5*x*A(5*x)^2) +...
simplifies to a power series in x with integer coefficients.

Cf. A218672, A218674, A218675, A218676.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^2,x,k*x)^k/k!*exp(-k*x*subst(A^2,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A218674 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(3n)/n! exp(-nxA(n*x)^3).

Original entry on oeis.org

1, 1, 4, 34, 455, 8710, 230077, 8285224, 407456797, 27587687551, 2596034329278, 342275007167359, 63606742005546232, 16730509857101195808, 6246818082857455197662, 3317816101992338134691233, 2510420393373091580780786808, 2709148467943025007607468405672

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare to the LambertW identity:

Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).

			O.g.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 455*x^4 + 8710*x^5 + 230077*x^6 +...
where
A(x) = 1 + x*A(x)^3*exp(-x*A(x)^3) + 2^2*x^2*A(2*x)^6/2!*exp(-2*x*A(2*x)^3) + 3^3*x^3*A(3*x)^9/3!*exp(-3*x*A(3*x)^3) + 4^4*x^4*A(4*x)^12/4!*exp(-4*x*A(4*x)^3) + 5^5*x^5*A(5*x)^15/5!*exp(-5*x*A(5*x)^3) +...
simplifies to a power series in x with integer coefficients.

Cf. A218672, A218673, A218675, A218676.

Cf. A217900, A218670, A218667, A218668, A218669, A134055.

{a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,n,k^k*x^k*subst(A^3,x,k*x)^k/k!*exp(-k*x*subst(A^3,x,k*x)+x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

cp's OEIS Frontend

A220955 O.g.f.: Sum_{n>=0} (2*n+1)^(2*n+1) * exp(-(2*n+1)^2*x) * x^n / n!.

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A221214 O.g.f.: Sum_{n>=0} (3*n+1)^(3*n+1) * exp(-(3*n+1)^3*x) * x^n / n!.

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A222526 O.g.f.: Sum_{n>=0} (n^6)^n * exp(-n^6*x) * x^n / n!.

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A222527 O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7*x) * x^n / n!.

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A222528 O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8*x) * x^n / n!.

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A222529 O.g.f.: Sum_{n>=0} (n^9)^n * exp(-n^9*x) * x^n / n!.

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A220955 O.g.f.: Sum_{n>=0} (2n+1)^(2n+1) * exp(-(2n+1)^2x) * x^n / n!.

A221214 O.g.f.: Sum_{n>=0} (3n+1)^(3n+1) * exp(-(3n+1)^3x) * x^n / n!.

A222526 O.g.f.: Sum_{n>=0} (n^6)^n * exp(-n^6x) x^n / n!.

A222527 O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7x) x^n / n!.

A222528 O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8x) x^n / n!.

A222529 O.g.f.: Sum_{n>=0} (n^9)^n * exp(-n^9x) x^n / n!.

A222530 O.g.f.: Sum_{n>=1} (n^10)^n * exp(-n^10x) x^n / n!.

A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2x)) (1 + n^2*x)^n / n!.

A218673 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(2n)/n! exp(-nxA(n*x)^2).

A218674 O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^nA(nx)^(3n)/n! exp(-nxA(n*x)^3).