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 1-8 of 8 results.

A118395 Expansion of e.g.f. exp(x + x^3).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 481, 2731, 10417, 91225, 681121, 3493711, 33597961, 303321877, 1938378625, 20282865331, 211375647841, 1607008257841, 18157826367937, 212200671085975, 1860991143630841, 22560913203079021, 289933758771407521, 2869267483843753147, 37116733726117707025
Offset: 0

Views

Author

Paul D. Hanna, May 07 2006

Keywords

Comments

Equals row sums of triangle A118394.

Links

Crossrefs

Cf. A118394, A118396.
Cf. A047974, A190875, A190877.

Programs

  • Magma
    [n le 3 select 1 else Self(n-1) + 3*(n-2)*(n-3)*Self(n-3): n in [1..26]]; // Vincenzo Librandi, Aug 25 2015
  • Maple
    with(combstruct):seq(count(([S, {S=Set(Union(Z, Prod(Z, Z, Z)))}, labeled], size=n)), n=0..22); # Zerinvary Lajos, Mar 18 2008
  • Mathematica
    CoefficientList[Series[E^(x+x^3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)
T[n_, k_] := n!/(k!(n-3k)!);
a[n_] := Sum[T[n, k], {k, 0, Floor[n/3]}];
a /@ Range[0, 24] (* Jean-François Alcover, Nov 04 2020 *)
  • PARI
    a(n)=n!*polcoeff(exp(x+x^3+x*O(x^n)),n)
  • PARI
    N=33;  x='x+O('x^N);
egf=exp(x+x^3);
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */
  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n-2*k, k)/(n-2*k)!); \\ Seiichi Manyama, Feb 25 2022
  • Sage
    def a(n):
    if (n<3): return 1
    else: return a(n-1) + 3*(n-1)*(n-2)*a(n-3)
[a(n) for n in (0..25)] # G. C. Greubel, Feb 18 2021

Formula

E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + (1+x^2)/(k+1)/(1-x/(x+(1)/G(k+1) )), recursively defined continued fraction. - Sergei N. Gladkovskii, Feb 04 2013
a(n) ~ 3^(n/3-1/2) * n^(2*n/3) * exp((n/3)^(1/3)-2*n/3). - Vaclav Kotesovec, Jun 02 2013
E.g.f.: A(x) = exp(x+x^3) satisfies A' - (1+3*x^2)*A = 0. - Gheorghe Coserea, Aug 24 2015
a(n+1) = a(n) + 3*n*(n-1)*a(n-2). - Gheorghe Coserea, Aug 24 2015
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,k)/(n-2*k)!. - Seiichi Manyama, Feb 25 2022

Extensions

Missing a(0)=1 prepended by Joerg Arndt, Sep 15 2012

A190877 Expansion of e.g.f. exp(x+x^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 15121, 1844641, 20013841, 119845441, 519072841, 1816454641, 223394731561, 3501661887361, 29675906201761, 177923109591361, 844925253766561, 104750282797418881
Offset: 0

Views

Author

Vladimir Kruchinin, May 23 2011

Keywords

Links

Crossrefs

Cf. A047974, A118395, A190875.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x+x^5],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 25 2015 *)
  • Maxima
    a(n):=n!*sum(binomial(n+(-4)*j,j)/(n+(-4)*j)!,j,0,n/4);
  • PARI
    a(n) = if(n<5, 1, a(n-1)+5!*binomial(n-1, 4)*a(n-5)); \\ Seiichi Manyama, Feb 25 2022

Formula

a(n) = n! * Sum_{j=0..n/4} binomial(n+(-4)*j,j)/(n+(-4)*j)!.
a(n) = a(n-1) + 5! * binomial(n-1,4) * a(n-5) for n > 4. - Seiichi Manyama, Feb 25 2022

A373522 Expansion of e.g.f. exp(x * (1 + x^3)^(1/3)).

Original entry on oeis.org

1, 1, 1, 1, 9, 41, 121, -279, -1679, 1009, 259281, 1173041, 669241, -267141159, -1295686391, 10821721, 650092657761, 3480768830561, 17723446561, -2911516748764191, -17068971040559639, 427036022281, 21673592659354854681, 137752098937383025481
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190875, A373523.
Cf. A373517.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n/3-k, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n/3-k,k)/(n-3*k)!.
a(n) == 1 mod 8.

A373523 Expansion of e.g.f. exp(x * (1 + x^3)^(2/3)).

Original entry on oeis.org

1, 1, 1, 1, 17, 81, 241, 1, 5601, 62497, 518561, 313281, 3999601, -40669199, 2177551377, 7318933441, 397613245121, -1411251083199, 9245424513601, -1554110065897343, 8222970963680721, 2117868896399761, 11780583339147607601, -55331596875625839999
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190875, A373522.
Cf. A373518.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(2*n/3-2*k, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*n/3-2*k,k)/(n-3*k)!.
a(n) == 1 mod 16.

A351905 Expansion of e.g.f. exp(x * (1 - x^3)).

Original entry on oeis.org

1, 1, 1, 1, -23, -119, -359, -839, 18481, 178417, 902161, 3318481, -69866279, -1011908039, -7204341143, -36194591159, 726745175521, 14326789219681, 131901636673441, 840736509931297, -16060449291985079, -408041402342457239, -4618341644958693959, -35691963052019431079
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2022

Keywords

Crossrefs

Cf. A293604, A246607, A351906.
Cf. A190875, A293493.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x^3))))
  • PARI
    a(n) = n!*sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)/(n-3*k)!);
  • PARI
    a(n) = if(n<4, 1, a(n-1)-4!*binomial(n-1, 3)*a(n-4));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k)/(n-3*k)!.
a(n) = a(n-1) - 4! * binomial(n-1,3) * a(n-4) for n > 3.

A351932 Number of set partitions of [n] such that block sizes are either 1 or 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 106, 442, 1786, 6106, 23596, 120836, 631632, 2854216, 13590396, 81258556, 510768316, 2839808572, 16008902296, 108643656136, 787965516416, 5161270717296, 33513036683512, 253407796702776, 2065728484459576, 15485032349429176, 113510664648701776
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A000085, A190865, A275423, A275425.
Cf. A190452, A190875, A351930.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
     `if`(n<4, 0, a(n-4)*binomial(n-1, 3))+a(n-1))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Feb 26 2022
seq(round(evalf(hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3))),n=0..28);  # Karol A. Penson, Jul 28 2023
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x+x^4/4!)))
  • PARI
    a(n) = n!*sum(k=0, n\4, 1/4!^k*binomial(n-3*k, k)/(n-3*k)!);
  • PARI
    a(n) = if(n<4, 1, a(n-1)+binomial(n-1, 3)*a(n-4));

Formula

E.g.f.: exp(x + x^4/24).
a(n) = n! * Sum_{k=0..floor(n/4)} (1/24)^k * binomial(n-3*k,k)/(n-3*k)!.
a(n) = a(n-1) + binomial(n-1,3) * a(n-4) for n > 3.
a(n) = hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3), Karol A. Penson, Jul 28 2023.

A373707 Expansion of e.g.f. exp(x * (1 + x^3)^2).

Original entry on oeis.org

1, 1, 1, 1, 49, 241, 721, 6721, 124321, 913249, 4243681, 94818241, 1640604241, 14642181841, 131026944049, 3669304504321, 62536989802561, 627395160826561, 10818406189690561, 308036857749752449, 5219006583104930161, 65146235714284117681
Offset: 0

Views

Author

Seiichi Manyama, Jun 14 2024

Keywords

Crossrefs

Cf. A361278, A373578, A373708.
Cf. A190875, A373522, A373523.
Cf. A373680.

Programs

  • PARI
    a(n) = n!*sum(k=0, 2*n\7, binomial(2*n-6*k, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/7)} binomial(2*n-6*k,k)/(n-3*k)!.
a(n) == 1 (mod 48).
a(n) = a(n-1) + 8*(n-1)*(n-2)*(n-3)*a(n-4) + 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-7).

A376565 E.g.f. satisfies A(x) = exp( x*A(x) * (1 + x^3*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 16, 149, 2016, 34447, 692224, 15986889, 420544000, 12494098331, 414681513984, 15201740343517, 609446038061056, 26511336043734375, 1243650774790045696, 62591481040666342673, 3364694927903114919936, 192423068815578523022899, 11665229364232192000000000
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2024

Keywords

Links

Crossrefs

Cf. A088695, A376564.
Cf. A190875, A376477.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, (n+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);

Formula

E.g.f.: (1/x) * Series_Reversion( x*exp(-x * (1 + x^3)) ).
a(n) = n! * Sum_{k=0..floor(n/4)} (n+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.
Showing 1-8 of 8 results.

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