A135746 -id:A135746 - cp's OEIS frontend

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

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).

A135744 E.g.f.: A(x) = Sum_{n>=0} exp( n(n+1)x ) * x^n/n!.

Original entry on oeis.org

1, 1, 5, 31, 297, 3781, 60373, 1188699, 28003825, 772499593, 24613769061, 894386029879, 36653691106585, 1679283513161229, 85360981759418485, 4781873479864452211, 293487961919565420897, 19633825959679051986961

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, A135745, A135746, A135747.

Flatten[{1, Table[Sum[Binomial[n, k]*(2*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))^(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

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

A355473 Expansion of Sum_{k>=0} x^k/(1 - k^3 * x)^(k+1).

Original entry on oeis.org

1, 1, 3, 28, 497, 12736, 517297, 28793248, 2095968065, 199522773568, 23839495688321, 3482169003693304, 616298415199306369, 130134007837039167040, 32272959284595295173377, 9313050358489324003967176, 3101245112865402456422252033

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..235

Cf. A000248, A135746, A355464.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^3*x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(1+sum(k=1, N, exp(k^3*x)*x^k/k!)))

a(n) = sum(k=0, n, k^(3*(n-k))*binomial(n, k));

E.g.f.: Sum_{k>=0} exp(k^3 * x) * x^k/k!.

a(n) = Sum_{k=0..n} k^(3*(n-k)) * binomial(n,k).

A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

Original entry on oeis.org

1, 2, 4, 17, 210, 9217, 1399298, 811229225, 2071392232962, 20710319937493889, 1137259214532706572162, 255141201504146525745627265, 348787971214016591166179037803522, 2262996819897931095524655885144485185409

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A086331, A320287, A349893, A355440, A355463.

Cf. A000248, A135746, A355473.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^k*x)^(k+1)))

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(k^k*x)*x^k/k!)))

a(n) = sum(k=0, n, k^(k*(n-k))*binomial(n, k));

E.g.f.: Sum_{k>=0} exp(k^k * x) * x^k/k!.

a(n) = Sum_{k=0..n} k^(k*(n-k)) * binomial(n,k).

A355471 Expansion of Sum_{k>=0} (x/(1 - k^2 * x))^k.

Original entry on oeis.org

1, 1, 2, 10, 77, 808, 11257, 196072, 4136897, 103755904, 3034193921, 101901347944, 3885951145969, 166605168800704, 7961498177012993, 420976047757358776, 24475992585921169553, 1556007778666449968128, 107625967130820901112833

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A080108, A135746, A234568, A355463, A355472.

Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(2*(n-k)), {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Feb 16 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^2*x))^k))

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A135744 E.g.f.: A(x) = Sum_{n>=0} exp( n*(n+1)*x ) * x^n/n!.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A355473 Expansion of Sum_{k>=0} x^k/(1 - k^3 * x)^(k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

Views

Author

Keywords

Crossrefs

Programs

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!.

A135744 E.g.f.: A(x) = Sum_{n>=0} exp( n(n+1)x ) * 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!.