A217913 -id:A217913 - cp's OEIS frontend

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

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

A218142 a(n) = Stirling2(n^2+n, n).

Original entry on oeis.org

1, 1, 31, 86526, 45232115901, 7713000216608565075, 666480349285726891499539272955, 41929298560838945526242744414099901692285884, 2610516895723221966171633379256064857587637240616032299710417

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

			O.g.f.: A(x) = 1 + x + 31*x^2 + 86526*x^3 + 45232115901*x^4 +...

Paul D. Hanna, Table of n, a(n) for n = 0..25

Cf. A008277, A218141, A218143, A007820, A217913, A217914, A217915.

Table[StirlingS2[n^2+n, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)

makelist(stirling2(n^2+n,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */

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

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n^2))), n^2)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n^2+n, n)}
for(n=0, 10, print1(a(n), ", "))

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

A218143 a(n) = Stirling2(n*(n+1)/2, n).

Original entry on oeis.org

1, 1, 3, 90, 34105, 210766920, 26585679462804, 82892803728383735268, 7529580759157036060608585183, 22982258052528294182955639980819773510, 2672446997421818663856559987803834697952486978300, 13239043631590111512460321918828937597837325561187113535696980

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 90*x^3 + 34105*x^4 + 210766920*x^5 + 26585679462804*x^6 +...

Paul D. Hanna, Table of n, a(n) for n = 0..30

Cf. A008277, A218141, A218142, A007820, A217913, A217914, A217915.

Table[StirlingS2[n*(n+1)/2, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)

makelist(stirling2(n*(n+1)/2,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n*(n-1)/2))), n*(n-1)/2)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n*(n+1)/2, n)}
for(n=0, 15, print1(a(n), ", "))

a(n) = [x^(n*(n-1)/2)] 1 / Product_{k=1..n} (1-k*x).

a(n) ~ n^(n*(n+1)/2)/n!. - Vaclav Kotesovec, May 11 2014

A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

Original entry on oeis.org

1, 15, 966, 145750, 40075035, 17505749898, 11143554045652, 9741955019900400, 11201516780955125625, 16392038075086211019625, 29749840488672593296243236, 65580126734167548918100615020, 172597131674172062132363512309613, 534584200037719212882636004559739000, 1924887533450780657560944228447179522880, 7973126100358260458973226689851075932667520, 37645241791600906804871080818625037726247519045

Offset: 1

Paul D. Hanna, Jul 03 2016

nonn

a(n) is divisible by the triangular numbers: a(n) / (n*(n+1)/2) = A274712(n).

			O.g.f.: A(x) = x + 15*x^2 + 966*x^3 + 145750*x^4 + 40075035*x^5 + 17505749898*x^6 + 11143554045652*x^7 + 9741955019900400*x^8 +...
where
A(x) = exp(-x)*x + 2^5*exp(-2^3*x)*x^2/2! + 3^8*exp(-3^3*x)*x^3/3! + 4^11*exp(-4^3*x)*x^4/4! + 5^14*exp(-5^3*x)*x^5/5! + 6^17*exp(-6^3*x)*x^6/6! + 7^20*exp(-7^3*x)*x^7/7! + 8^23*exp(-8^3*x)*x^8/8! +...+ n^(3*n-1)*exp(-n^3*x)*x^n/n! +...
simplifies to an integer series.

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A129506, A217913, A274712, A008277.

Table[StirlingS2[3*n - 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Jul 06 2016 *)

{a(n) = abs( stirling(3*n-1, n, 2) )}
for(n=1, 20, print1(a(n), ", "))

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

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

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

O.g.f.: Sum_{n>=1} n^(3*n-1) * exp(-n^3*x) * x^n / n!, an integer series.
a(n) = A008277(3*n-1,n) for n>=1, where A008277 are the Stirling numbers of the second kind.
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(3*n-1).
a(n) = [x^(2*n-1)] 1 / Product_{k=1..n} (1 - k*x).
a(n) ~ 3^(3*n-1) * n^(2*n-3/2) / (exp(2*n) * c^n * (3-c)^(2*n-1) * sqrt(2*Pi*(1-c))), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968... = -A226750. - Vaclav Kotesovec, Jul 06 2016

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

Original entry on oeis.org

1, 6, 266, 22275, 2757118, 452329200, 92484925445, 22653141490980, 6466506598695390, 2108114165258886708, 772778072287000494520, 314641228029527540596455, 140880584836935832288402135, 68799366730032076856334789900, 36392216443342587869022660451080, 20728132932716479897744043460870000

Offset: 0

Vaclav Kotesovec, May 13 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A007820, A348084, A383882.

Cf. A217913.

[&+[Abs(StirlingSecond(4*n, 3*n))]: n in [0..15]]; // Vincenzo Librandi, May 21 2025

Table[SeriesCoefficient[Product[1/(1-k*x), {k, 1, 3*n}], {x, 0, n}], {n, 0, 15}]
Table[StirlingS2[4*n, 3*n], {n, 0, 15}]
Table[SeriesCoefficient[(-1)^n/(Pochhammer[1 - 1/x, 3*n]*x^(3*n)), {x, 0, n}], {n, 0, 15}]

a(n) = Stirling2(4*n,3*n).

a(n) ~ (-1)^(3*n) * 4^(4*n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + w)) * exp(n) * 3^(3*n + 1/2) * w^(3*n) * (4/3 + w)^n), where w = LambertW(-4/(3*exp(4/3))).

A217912 O.g.f.: Sum_{n>=0} 2n^n (3n+2)^(n-1) exp(-n(3n+2)x) x^n / n!.

Original entry on oeis.org

1, 2, 22, 602, 26656, 1643054, 130318966, 12666846728, 1459524093232, 194626267782398, 29495119281572770, 5008297010070635978, 942044179147597185544, 194462342099815302424136, 43711609296992502659474632, 10628894996508864880841838416, 2780041837527932453797746700384

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} 2*(3*n+2)^(n-1) * exp(-(3*n+2)*x) * x^n/n!.

			O.g.f.: A(x) = 1 + 2*x + 22*x^2 + 602*x^3 + 26656*x^4 + 1643054*x^5 + ...
where
A(x) = 1 + 2*1^1*5^0*x*exp(-1*5*x) + 2*2^2*8^1*exp(-2*8*x)*x^2/2! + 2*3^3*11^2*exp(-3*11*x)*x^3/3! + 2*4^4*14^3*exp(-4*14*x)*x^4/4! + 2*5^5*17^4*exp(-5*17*x)*x^5/5! + ...
simplifies to a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A217900, A217910, A217911, A217913.

Flatten[{1,Table[Sum[Binomial[n-1,j]*3^j*2^(n-j)*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)

makelist( if n=0 then 1 else 1/n! * sum(2*(-1)^(n-k)*binomial(n,k) * k^n * (3*k+2)^(n-1),k,0,n), n, 0, 30); /* Martin Ettl, Oct 15 2012 */

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

{a(n)=1/n!*polcoeff(sum(k=0, n, 2*k^k*(3*k+2)^(k-1)*x^k/(1+k*(3*k+2)*x +x*O(x^n))^(k+1)), n)}

{a(n)=1/n!*sum(k=0,n, 2*(-1)^(n-k)*binomial(n,k)*k^n*(3*k+2)^(n-1))}
for(n=0,30,print1(a(n),", "))

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

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

a(n) = 1/n! * Sum_{k=0..n} 2*(-1)^(n-k)*binomial(n,k) * k^n * (3*k+2)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} 2*k^k*(3*k+2)^(k-1)*x^k / (1 + k*(3*k+2)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+2*x)^(n-1) / Product_{k=1..n} (1 - 3*k*x).
a(n) = [x^n] 1 + x*(1-2*x)^(n-1) / Product_{k=1..n} (1 - (3*k+2)*x).
a(n) ~ 2^(2*n-1/6) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/3)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

A219228 O.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * (n^3x)^n/n! exp(-n^3xA(x)).

Original entry on oeis.org

1, 1, 32, 3119, 625710, 214333471, 112105268136, 83149960883200, 83014425998481126, 107334569041127441462, 174471878478682785998864, 348242875992753988109552778, 837327855535084109106340786272, 2387108242583316451939303856237037

Offset: 0

Paul D. Hanna, Nov 15 2012

nonn

Compare to the o.g.f. of A217913:

Sum_{n>=0} (n^3)^n*exp(-n^3*x)*x^n/n! = Sum_{n>=0} Stirling2(3*n,n)*x^n.

			O.g.f.: A(x) = 1 + x + 32*x^2 + 3119*x^3 + 625710*x^4 + 214333471*x^5 +...
where
A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^6*x^2*A(x)^2/2!*exp(-8*x*A(x)) + 3^9*x^3*A(x)^3/3!*exp(-27*x*A(x)) + 4^12*x^4*A(x)^4/4!*exp(-64*x*A(x)) + 5^15*x^5*A(x)^5/5!*exp(-125*x*A(x)) +...
simplifies to a power series in x with integer coefficients.
G.f. A(x) satisfies A(x) = G(x*A(x)) where G(x) = A(x/G(x)) begins:
G(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + 210766920*x^5 + 110687251039*x^6 +...+ Stirling2(3*n,n)*x^n +...

Cf. A217913, A219184, A217900, A218681, A218672.

{a(n)=local(A=1);for(i=1,n,A=sum(m=0, n, (m^3*x*A)^m/m!*exp(-m^3*x*A+x*O(x^n))));polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

O.g.f. satisfies: A(x) = Sum_{n>=0} Stirling2(3*n,n) * x^n * A(x)^n.

A222053 O.g.f.: Sum_{n>=0} (n^3x)^n/(1-n^3x)^n * exp(-n^3x/(1-n^3x)) / n!.

Original entry on oeis.org

1, 1, 32, 3536, 877221, 394506859, 284110844070, 302350295364613, 449340338669205876, 894210483750815778132, 2306748823711254973903838, 7516588630649080782251419791, 30292392269310179039574629318038, 148358895760995636729844370111255773

Offset: 0

Paul D. Hanna, Mar 08 2013

nonn

			O.g.f.: A(x) = 1 + x + 32*x^2 + 3536*x^3 + 877221*x^4 + 394506859*x^5 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^6*x^2/(1-2^3*x)^2*exp(-2^3*x/(1-2^3*x))/2! + 3^9*x^3/(1-3^3*x)^3*exp(-3^3*x/(1-3^3*x))/3! + 4^12*x^4/(1-4^3*x)^4*exp(-4^3*x/(1-4^3*x))/4! +...
simplifies to a power series in x with integer coefficients.

Cf. A134055, A174845, A222054, A217913, A008277.

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

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(3*n, k)))}
for(n=0, 25, print1(a(n), ", "))

a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(3*n,k) for n>0 with a(0)=1.

A222054 O.g.f.: Sum_{n>=0} (n^4x)^n/(1-n^4x)^n * exp(-n^4x/(1-n^4x)) / n!.

Original entry on oeis.org

1, 1, 128, 90621, 193322261, 933620289929, 8632521193856886, 136885314823146617517, 3443427946573913689696192, 129667338445150206244162849650, 6988095504452769015520539806767120, 520011535068804196524689647521015780176

Offset: 0

Paul D. Hanna, Mar 08 2013

nonn

			O.g.f.: A(x) = 1 + x + 128*x^2 + 90621*x^3 + 193322261*x^4 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^8*x^2/(1-2^4*x)^2*exp(-2^4*x/(1-2^4*x))/2! + 3^12*x^3/(1-3^4*x)^3*exp(-3^4*x/(1-3^4*x))/3! + 4^16*x^4/(1-4^4*x)^4*exp(-4^4*x/(1-4^4*x))/4! +...
simplifies to a power series in x with integer coefficients.

Cf. A134055, A174845, A222053, A217913, A008277.

{a(n)=polcoeff(sum(k=0, n+1, (k^4*x)^k/(1-k^4*x)^k*exp(-k^4*x/(1-k^4*x+x*O(x^n)))/k!), n)}
for(n=0, 20, print1(a(n), ", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(4*n, k)))}
for(n=0, 20, print1(a(n), ", "))

a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(4*n,k) for n>0 with a(0)=1.

cp's OEIS Frontend

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

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

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Mathematica

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A218142 a(n) = Stirling2(n^2+n, n).

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Maxima

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A218143 a(n) = Stirling2(n*(n+1)/2, n).

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Maxima

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A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

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A383881 a(n) = [x^n] Product_{k=1..3*n} 1/(1 - k*x).

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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!.

A383881 a(n) = [x^n] Product_{k=1..3n} 1/(1 - kx).

A217912 O.g.f.: Sum_{n>=0} 2n^n (3n+2)^(n-1) exp(-n(3n+2)x) x^n / n!.

A219228 O.g.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * (n^3x)^n/n! exp(-n^3xA(x)).

A222053 O.g.f.: Sum_{n>=0} (n^3x)^n/(1-n^3x)^n * exp(-n^3x/(1-n^3x)) / n!.

A222054 O.g.f.: Sum_{n>=0} (n^4x)^n/(1-n^4x)^n * exp(-n^4x/(1-n^4x)) / n!.