A135744 -id:A135744 - cp's OEIS frontend

A135746 E.g.f.: A(x) = Sum_{n>=0} exp(n^2x) x^n/n!.

1, 1, 3, 16, 137, 1536, 22417, 407884, 8920641, 230576320, 6928080641, 238375169484, 9288784476193, 406150114297552, 19761959813464065, 1062437048084297596, 62727815353861478273, 4045278841893314992896

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1536*x^5/5! + ...
where A(x) = 1 + exp(x)*x + exp(4*x)*x^2/2! + exp(9*x)*x^3/3! + exp(16*x)*x^4/4! + exp(25*x)*x^5/5! + ...
O.g.f.: F(x) = 1 + x + 3*x^2 + 16*x^3 + 137*x^4 + 1536*x^5 + 22417*x^6 + ...
where F(x) = 1 + x/(1-x)^2 + x^2/(1-4*x)^3 + x^3/(1-9*x)^4 + x^4/(1-16*x)^5 + x^5/(1-25*x)^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. variants: A135742, A135743, A135744, A135745, A135747.

Cf. A000248.

Flatten[{1, Table[Sum[Binomial[n, k]*k^(2*(n - k)), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)

{a(n)=sum(k=0,n,binomial(n,k)*(k^2)^(n-k))}

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

{a(n)=polcoeff(sum(k=0, n, x^k/(1-k^2*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Aug 08 2009

a(n) = Sum_{k=0..n} C(n,k)*(k^2)^(n-k).
O.g.f.: Sum_{n>=0} x^n/(1 - n^2*x)^(n+1). - Paul D. Hanna, Aug 08 2009
a(n) ~ n^(n + 1/2) * r^(2*n - 3*r + 1/2) / (sqrt(2*n + 3*r) * (n - r)^(n - r + 1/2)), where r = (n/w) * (1 + (w-1)/((2*w^2 + w - 2)/log(w-1) - w + 2)) and w = LambertW(exp(1)*n). - Vaclav Kotesovec, Jul 05 2022

A135742 E.g.f.: A(x) = Sum_{n>=0} exp( n(n-1)/2 x ) * x^n / n!.

Original entry on oeis.org

1, 1, 1, 4, 19, 131, 1156, 12622, 166825, 2600677, 47038456, 974165336, 22829939089, 599668759483, 17512623094240, 564613124026876, 19972670155565761, 771019774737952313, 32326390781950804048

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. variants: A135743, A135744, A135745, A135746.

Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[k, 2]^(n - k), {k, 0, n}], {n,1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)

{a(n)=sum(k=0,n,binomial(n,k)*(k*(k-1)/2)^(n-k))}
for(n=0,25,print1(a(n),", "))

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

/* From Sum_{n>=0} x^n/(1 - n*(n-1)/2*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k-1)/2*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))

a(n) = Sum_{k=0..n} C(n,k) * ( k*(k-1)/2 )^(n-k).

O.g.f.: Sum_{n>=0} x^n / (1 - n*(n-1)/2 * x)^(n+1). - Paul D. Hanna, Jul 30 2014

A135743 E.g.f.: A(x) = Sum_{n>=0} exp(n(n+1)/2x)*x^n/n!.

Original entry on oeis.org

1, 1, 3, 13, 83, 686, 7132, 90343, 1357449, 23783068, 478784096, 10938189329, 280771780489, 8029138915630, 253911056912892, 8823070442039641, 335009138739028673, 13830540214264709000, 618085473234055115968

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

			E.g.f.: 1 + x + 3*x^2/2! + 13*x^3/3! + 83*x^4/4! +...
= 1 + exp(x)*x + exp(3x)*x^2/2! + exp(6x)*x^3/3! + exp(10x)*x^4/4! +...
O.g.f.: 1 + x + 3*x^2 + 13*x^3 + 83*x^4 + 686*x^5 +...
= 1 + x/(1-x)^2 + x^2/(1-3x)^3 + x^3/(1-6x)^4 + x^4/(1-10x)^5 +...

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. variants: A135742, A135744, A135745, A135746.

Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[k + 1, 2]^(n - k), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)

{a(n)=sum(k=0,n,binomial(n,k)*(k*(k+1)/2)^(n-k))}

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

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

a(n) = Sum_{k=0..n} C(n,k)*[k*(k+1)/2]^(n-k).

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

A135745 E.g.f.: A(x) = Sum_{n>=0} exp((n-1)x)^n x^n/n!.

Original entry on oeis.org

1, 1, 1, 7, 49, 501, 6841, 115123, 2362305, 57768553, 1646192881, 53952383871, 2010872281969, 84330050952733, 3945169959883881, 204416253047774251, 11655594262050124801, 727189793270478477777, 49395902623624761264865

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. variants: A135742, A135743, A135744, A135746, A135747.

Flatten[{1, Table[Sum[Binomial[n, k]*(k*(k - 1))^(n - k), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)

{a(n)=sum(k=0,n,binomial(n,k)*(k*(k-1))^(n-k))}
for(n=0,25,print1(a(n),", "))

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

/* From Sum_{n>=0} x^n/(1 - n*(n-1)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k-1)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))

a(n) = Sum_{k=0..n} C(n,k)*[k*(k-1)]^(n-k).

O.g.f.: Sum_{n>=0} x^n/(1 - n*(n-1)*x)^(n+1). - Paul D. Hanna, Jul 30 2014

A135747 E.g.f.: A(x) = Sum_{n>=0} exp( (n^2-1)x ) x^n/n!.

Original entry on oeis.org

1, 0, 2, 9, 88, 985, 14976, 278929, 6208000, 163268865, 4979147680, 173500986241, 6838921208736, 302161792811905, 14840867887070512, 804732692174218305, 47888731015720316416, 3110871265807567331329, 219546952410733092279360

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

n divides a(n) for n>=1.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. variants: A135742, A135743, A135744, A135745, A135746.

Flatten[{1, Table[Sum[Binomial[n, k]*(k^2 - 1)^(n - k), {k, 0, n}], {n,1,25}]}] (* G. C. Greubel, Nov 05 2016 *)

{a(n)=sum(k=0,n,binomial(n,k)*(k^2-1)^(n-k))}
for(n=0,25,print1(a(n),", "))

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

/* From Sum_{n>=0} x^n/(1 - (n^2-1)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(k^2-1)*x +x*O(x^n))^(k+1)), n)}
for(n=0,25,print1(a(n),", "))

a(n) = Sum_{k=0..n} C(n,k) * (k^2-1)^(n-k).

O.g.f.: Sum_{n>=0} x^n / (1 - (n^2-1)*x)^(n+1). - Paul D. Hanna, Jul 30 2014

cp's OEIS Frontend

A135746 E.g.f.: A(x) = Sum_{n>=0} exp(n^2*x) * x^n/n!.

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A135742 E.g.f.: A(x) = Sum_{n>=0} exp( n*(n-1)/2 * x ) * x^n / n!.

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A135743 E.g.f.: A(x) = Sum_{n>=0} exp(n*(n+1)/2*x)*x^n/n!.

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A135745 E.g.f.: A(x) = Sum_{n>=0} exp((n-1)*x)^n * x^n/n!.

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A135747 E.g.f.: A(x) = Sum_{n>=0} exp( (n^2-1)*x ) * x^n/n!.

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A135746 E.g.f.: A(x) = Sum_{n>=0} exp(n^2x) x^n/n!.

A135742 E.g.f.: A(x) = Sum_{n>=0} exp( n(n-1)/2 x ) * x^n / n!.

A135743 E.g.f.: A(x) = Sum_{n>=0} exp(n(n+1)/2x)*x^n/n!.

A135745 E.g.f.: A(x) = Sum_{n>=0} exp((n-1)x)^n x^n/n!.

A135747 E.g.f.: A(x) = Sum_{n>=0} exp( (n^2-1)x ) x^n/n!.