A222528 -id:A222528 - cp's OEIS frontend

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

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

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

cp's OEIS Frontend

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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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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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!.

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!.