cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A086331 Expansion of e.g.f. exp(x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 2, 7, 43, 393, 4721, 69853, 1225757, 24866481, 572410513, 14738647221, 419682895325, 13094075689225, 444198818128313, 16278315877572141, 640854237634448101, 26973655480577228769, 1208724395795734172705, 57453178877303382607717, 2887169565412587866031533
Offset: 0

Views

Author

Vladeta Jovovic, Sep 01 2003

Keywords

Comments

Binomial transform of A000312. - Tilman Neumann, Dec 13 2008
a(n) is the number of partial functions on {1,2,...,n} that are endofunctions. See comments in A000169 and A126285 by Franklin T. Adams-Watters. - Geoffrey Critzer, Dec 19 2011

Examples

			a(2) = 7 because {}->{}, 1->1, 2->2, and the four functions from {1,2} into {1,2}. Note A000169(2) = 9 because it counts these 7 and 1->2, 2->1.

Links

Crossrefs

Cf. A069856, A204042, A277454, A277456, A323280.

Programs

  • Maple
    a:= n-> add(binomial(n,k)*k^k, k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 30 2021
  • Mathematica
    nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[x]/(1-t),{x,0,nn}],x]  (* Geoffrey Critzer, Dec 19 2011 *)
  • PARI
    a(n) = sum(k=0,n, binomial(n, k)*k^k ); \\ Joerg Arndt, May 10 2013
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k*x)^k/k!))) \\ Seiichi Manyama, Jul 04 2022

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*k^k.
a(n) ~ e^(1/e)*n^n * (1 + 1/(2*e*n)) ~ 1.444667861... * n^n. - Vaclav Kotesovec, Nov 27 2012
G.f.: Sum_{k>=0} (k * x)^k/(1 - x)^(k+1). - Seiichi Manyama, Jul 04 2022

A308490 a(0) = 1, a(n) = Sum_{k=1..n} stirling2(n,k) * k^(2*k).

Original entry on oeis.org

1, 1, 17, 778, 70023, 10439451, 2327592658, 725325847443, 301054612941037, 160546901676583432, 106969402879501806589, 87079496403914056543799, 85043317211453886535179728, 98135961356804028347727824541, 132097548629285541942722646521053
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2019

Keywords

Links

Crossrefs

Cf. A229261, A282190, A308491, A316747, A323280, A351182.

Programs

  • Mathematica
    Join[{1}, Table[Sum[k^(2*k)*StirlingS2[n, k], {k, 1, n}], {n, 1, 20}]]
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*(exp(x)-1))^k/k!))) \\ Seiichi Manyama, Feb 04 2022

Formula

a(n) ~ exp(exp(-2)/2) * n^(2*n).
E.g.f.: Sum_{k>=0} (k^2 * (exp(x) - 1))^k / k!. - Seiichi Manyama, Feb 04 2022

A064570 Binomial transform of (2n)!.

Original entry on oeis.org

1, 3, 29, 799, 43353, 3837851, 501393973, 90608944119, 21633834338609, 6593857931708083, 2497877833687172301, 1151118261673522046543, 634098400947597342716809, 411445662820653995008883019
Offset: 0

Views

Author

Karol A. Penson, Sep 20 2001

Keywords

Comments

Compare with A229464. - Peter Bala, Sep 25 2013

Links

Crossrefs

Cf. A010050, A229464, A001813, A323280.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*(2*k)!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 26 2013 *)

Formula

In Maple notation: a(n)=hypergeom([1, 1/2, -n], [], -4), n=0, 1, ...
a(n) = Integral_{x>=0} ((x^4-1)/(x^2-1))^n*exp(-x) dx. - Gerald McGarvey, Oct 14 2006
From Peter Bala, Sep 25 2013: (Start)
a(n) = Sum_{k = 0..n} binomial(n,k)*(2*k)!.
Clearly a(n) is always odd; indeed, a(n) = 1 + 2*n*A229464(n-1) for n >= 1.
Recurrence equation: a(n) = 1 + 2*n*(2*n - 1)*a(n-1) - 2*n*(2*n - 2)*a(n-2) with a(0) = 1 and a(1) = 3.
O.g.f. Sum_{k >= 0} (2*k)!*x^k/(1 - x)^(k + 1) = 1 + 3*x + 29*x^2 + 799*x^3 + .... (End)
Recurrence (homogeneous): a(n) = (4*n^2 - 2*n + 1)*a(n-1) - 2*(n-1)*(4*n-3)*a(n-2) + 4*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Sep 26 2013
a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / exp(2*n). - Vaclav Kotesovec, Sep 26 2013
From Peter Bala, Nov 26 2017: (Start)
E.g.f.: exp(x)*Sum_{n >= 0} A001813(n)*x^n.
a(k) = a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
More generally a(n+k) = a(n) (mod k) for all n and k by an induction argument on n.
It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 3, 9, 9, 3, 1, 3, 9, 9, 3, ... with exact period 5. (End)

A355495 Expansion of Sum_{k>=0} (k^2 * x/(1 - x))^k.

Original entry on oeis.org

1, 1, 17, 762, 67772, 10032208, 2226273192, 691431572992, 286268594755712, 152365547943819264, 101361042063083269520, 82409537565402784477984, 80397802305461995791664944, 92692687015689239272783171264
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2022

Keywords

Links

Crossrefs

Cf. A355494, A355496.
Cf. A323280.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x/(1-x))^k))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, k^(2*k)*binomial(n-1, k-1)));

Formula

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

A355493 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - x)^(k+1).

Original entry on oeis.org

1, 2, 67, 19879, 16856337, 30601661681, 101743314190033, 559257425236996361, 4726837695171258085569, 58192258417571877186113281, 1000581709943568968705788233921, 23236157618902718144948494353385025, 709080642850925779233576351761544968833
Offset: 0

Views

Author

Seiichi Manyama, Jul 04 2022

Keywords

Links

Crossrefs

Cf. A086331, A323280.
Cf. A355470, A355473, A355496.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^3*x)^k/(1-x)^(k+1)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k^3*x)^k/k!)))
  • PARI
    a(n) = sum(k=0, n, k^(3*k)*binomial(n, k));

Formula

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

A360647 Expansion of Sum_{k>=0} (k^2 * x * (1 + x))^k.

Original entry on oeis.org

1, 1, 17, 761, 67739, 10029956, 2226004406, 691381685259, 286255287677425, 152360721379689043, 101358756787489940837, 82408168580060017122144, 80396790074312939684672316, 92691781529853274368541343021
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2023

Keywords

Crossrefs

Cf. A323280, A360611.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[Binomial[n-k,k] * (n-k)^(2*(n-k)), {k,0,n/2}], {n,1,30}]}] (* Vaclav Kotesovec, Feb 16 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x*(1+x))^k))
  • PARI
    a(n) = sum(k=0, n\2, (n-k)^(2*(n-k))*binomial(n-k, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (n-k)^(2*(n-k)) * binomial(n-k,k).
a(n) ~ n^(2*n). - Vaclav Kotesovec, Feb 16 2023
Showing 1-6 of 6 results.

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