A007820 -id:A007820 - 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

A247238 a(n) = Stirling2(2*n+1, n).

Original entry on oeis.org

1, 15, 301, 7770, 246730, 9321312, 408741333, 20415995028, 1144614626805, 71187132291275, 4864251308951100, 362262620784874680, 29206898819153109600, 2534474684137526739000, 235535731151727520125765, 23339590705557273894321960

Offset: 1

Vladimir Kruchinin, Nov 28 2014

nonn

			O.g.f.: A(x) = x + 15*x^2 + 301*x^3 + 7770*x^4 + 246730*x^5 + 9321312*x^6 + ... where A(x) = 1^3*x*exp(-1^2*x) + 2^5*exp(-2^2*x)*x^2/2! + 3^7*exp(-3^2*x)*x^3/3! + 4^9*exp(-4^2*x)*x^4/4! + 5^11*exp(-5^2*x)*x^5/5! + ...

G. C. Greubel, Table of n, a(n) for n = 1..340

Cf. A048993, A007820, A217899, A243227, A226775.

Table[StirlingS2[2*n+1, n], {n, 1, 20}] (* Vaclav Kotesovec, Nov 29 2014 *)

vector(50, n, stirling(2*n+1, n, 2)) \\ Colin Barker, Nov 28 2014

a(n) = A243227(n) / (n-1)!. - Vaclav Kotesovec, Nov 29 2014
a(n) ~ 2^(2*n+1/2) * n^(n+1/2) / (sqrt(Pi) * sqrt(1-c) * exp(n) * c^n * (2-c)^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775). - Vaclav Kotesovec, Nov 29 2014
O.g.f. Sum_{n>=1} n^(2*n+1) * x^n * exp(-n^2*x) / n! = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Oct 09 2023

A348084 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k x).

Original entry on oeis.org

1, 3, 65, 2646, 159027, 12662650, 1256328866, 149304004500, 20677182465555, 3270191625210510, 581535955088511150, 114860168436414644100, 24947615188488584876910, 5909902157669174347277556, 1516413100622001261250104100, 418965782063742792530650053000

Offset: 0

Seiichi Manyama, Sep 28 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..300

Cf. A007820 (S2(2*n,n)), A008277 (S2(n,k)), A129505, A348085.

a(n) = polcoef(1/prod(k=1, 2*n, 1-k*x+x*O(x^n)), n);

PARI
```
a(n) = stirling(3*n, 2*n, 2);
```

a(n) = sum(k=0, 2*n, (-1)^k*k^(3*n)*binomial(2*n, k))/(2*n)!;

from sympy.functions.combinatorial.numbers import stirling
def A348084(n): return stirling(3*n,2*n) # Chai Wah Wu, Jun 09 2025

a(n) = S2(3*n,2*n), where S2(n,k) is Stirling numbers of the second kind.
a(n) = (1/(2*n)!) * Sum_{k=0..2*n} (-1)^k * k^(3*n) * binomial(2*n,k).
a(n) ~ 3^(3*n) * n^(n - 1/2) / (sqrt(Pi*(1-c)) * 2^(n+1) * exp(n) * (3 - 2*c)^n * c^(2*n)), where c = -LambertW(-3*exp(-3/2)/2) = 0.62578253420128292093838... - Vaclav Kotesovec, Oct 02 2021

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!.

A348084 a(n) = [x^n] Product_{k=1..2n} 1/(1 - k x).