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 1-5 of 5 results.

A381817 Expansion of (1/x) * Series_Reversion( x * (1-x) / C(x) ), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 8, 41, 239, 1507, 10016, 69123, 490676, 3560150, 26285896, 196862679, 1491921261, 11420072162, 88166571504, 685724643699, 5367842153463, 42259058503891, 334373741310812, 2657683458672907, 21209720057079565, 169886023881795700, 1365290865904393560
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A381818, A381819, A381820.
Cf. A000108.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*2*x/(1-sqrt(1-4*x)))/x)

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(2*n-k,n-k)/(n+2*k+1).
D-finite with recurrence 270*n*(n-1)*(2*n+1)*(4886806261*n -12359738163)*(n+1)*a(n) +36*n*(n-1)*(73302093915*n^3 -4013759132354*n^2 +11228589268975*n -4731576382254)*a(n-1) -6*(n-1)*(78948725805818*n^4 -721014042837927*n^3 +2114039183987386*n^2 -2373558292742247*n +834825525358878)*a(n-2) +(3703469060597227*n^5 -40768871113864973*n^4 +173554734639707111*n^3 -360669855974794759*n^2 +370762762031723274*n -153683482287306096)*a(n-3) +6*(-2284895393144753*n^5 +28245013068548213*n^4 -138588666805096327*n^3 +341806596235129383*n^2 -433338949590369664*n +232825263110939100)*a(n-4) +10*(5*n-22)*(5*n-21) *(5*n-19)*(5*n-18)*(1032930487477*n -4077934418263)*a(n-5)=0. - R. J. Mathar, Mar 10 2025

A381819 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^3 ) )^(1/3), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 16, 177, 2271, 31731, 468614, 7195295, 113712012, 1837457589, 30220139048, 504212998955, 8513461623355, 145197727340337, 2497695979786842, 43285207907364178, 755005614380697735, 13244500528948104210, 233515959911770430972, 4135792046643993604967
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A381817, A381818, A381820.
Cf. A000108, A381773.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*2*x/(1-sqrt(1-4*x)))^3)/x)^(1/3))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)^3) / (1 - x*A(x)^3).
a(n) = Sum_{k=0..n} binomial(3*n+2*k+1,k) * binomial(4*n-k,n-k)/(3*n+2*k+1).

A381820 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) / C(x))^4 ) )^(1/4), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 20, 281, 4599, 82113, 1550993, 30473930, 616463800, 12753523628, 268586285058, 5738804673016, 124098812744140, 2710824280371114, 59728504549831296, 1325862161472193292, 29623682752417138511, 665679666998856945540, 15034747192791290846435
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A381817, A381818, A381819.
Cf. A000108, A381774.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec((serreverse(x*((1-x)*2*x/(1-sqrt(1-4*x)))^4)/x)^(1/4))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)^4) / (1 - x*A(x)^4).
a(n) = Sum_{k=0..n} binomial(4*n+2*k+1,k) * binomial(5*n-k,n-k)/(4*n+2*k+1).

A381831 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^3 ) )^(1/3).

Original entry on oeis.org

1, 2, 14, 133, 1456, 17306, 217066, 2827896, 37895130, 519000037, 7232429952, 102220846756, 1461817707558, 21112968248198, 307527937374182, 4512344039147420, 66634574697351360, 989569163283434676, 14769533757869187052, 221426909287107012800, 3333042591222552282784, 50353576994047154278451
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2025

Keywords

Crossrefs

Cf. A364592, A381818, A381830.
Cf. A129442, A381828.
Cf. A000108.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^3)/x)^(1/3))

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(3*n+k+1,k) * binomial(4*n-2*k,n-k)/(3*n+k+1).

A381830 G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 10, 69, 558, 4946, 46506, 455587, 4599494, 47517909, 499933964, 5337957532, 57694565830, 630010984557, 6939976239376, 77027050722166, 860564349616694, 9670164031087137, 109221767288604000, 1239281689627682221, 14119315749935075540, 161460732437631678114
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2025

Keywords

Crossrefs

Cf. A364592, A381818, A381831.
Cf. A000108, A381778.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(2*n, n-k)/(n+3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(2*n,n-k)/(n+3*k+1).
Showing 1-5 of 5 results.

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