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 1-3 of 3 results.

A361597 Expansion of e.g.f. exp( x^3/(6 * (1-x)^3) ) / (1-x).

Original entry on oeis.org

1, 1, 2, 7, 40, 320, 3130, 34930, 432320, 5866840, 86816800, 1395455600, 24270908200, 454897042600, 9146979842000, 196443726879400, 4486709145318400, 108548344109004800, 2771885136281060800, 74475606190225240000, 2099591224223100608000
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A002720, A361596.
Cf. A361573.

Programs

  • Mathematica
    Table[n! * Sum[Binomial[n,3*k]/(6^k * k!), {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3))/(1-x)))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/(6^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
Recurrence: 2*a(n) = 2*(4*n - 3)*a(n-1) - 6*(n-1)*(2*n - 3)*a(n-2) + (n-2)*(n-1)*(8*n - 17)*a(n-3) - 2*(n-3)^2*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(-7/8) * exp(-1/24 + 5*2^(-15/4)*n^(1/4)/3 - sqrt(n/2)/2 + 2^(7/4)*n^(3/4)/3 - n) * n^(n + 1/8) * (1 + (2637/10240)*2^(3/4)/n^(1/4)). (End)

A361577 Expansion of e.g.f. exp(x^4/(24 * (1 - x)^4)).

Original entry on oeis.org

1, 0, 0, 0, 1, 20, 300, 4200, 58835, 849240, 12814200, 203742000, 3430355775, 61363001700, 1168815948300, 23734579869000, 513878948207625, 11850279026586000, 290440507342986000, 7543064638441332000, 206860683821114948625, 5968372055889205462500
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2023

Keywords

Crossrefs

Cf. A000262, A335344, A361573.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/(24*(1-x)^4))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, (-1)^(j-4)*j*binomial(-4, j-4)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-1,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} (-1)^(k-4) * k * binomial(-4,k-4) * a(n-k)/(n-k)!.

A373757 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^2)).

Original entry on oeis.org

1, 0, 0, 1, 8, 60, 490, 4480, 45920, 524440, 6619200, 91568400, 1377884200, 22401579200, 391192401600, 7300174281400, 144938169376000, 3049711320656000, 67777255079934400, 1586172656920051200, 38984454900431040000, 1003827897443395024000
Offset: 0

Views

Author

Seiichi Manyama, Jun 17 2024

Keywords

Crossrefs

Cf. A361533, A361573.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x^3/(6(1-x)^2)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 30 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n-k-1, n-3*k)/(6^k*k!));
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, j*(j-2)*v[i-j+1]/(i-j)!)); v;

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-k-1,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * (k-2) * a(n-k)/(n-k)!.
Showing 1-3 of 3 results.

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