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 1-7 of 7 results.

A381772 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 11, 83, 727, 6940, 70058, 735502, 7949031, 87851819, 988307647, 11279719247, 130286197186, 1520108988221, 17889102534329, 212095541328931, 2531001870925559, 30376237591559863, 366417240105654587, 4440000077166319993, 54020150448778625847, 659665548217188211288
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A054727, A060941, A381773, A381774, A381775.
Cf. A007852, A069271, A381780.
Cf. A000108, A274052.

Programs

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

Formula

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

A381774 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, 19, 255, 3995, 68344, 1237526, 23316295, 452385355, 8977539540, 181374792040, 3718002102747, 77138798530854, 1616741658725930, 34179703551312530, 728019711835819493, 15608122038151106507, 336551042553481867640, 7293934071668996347055
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A054727, A060941, A381772, A381773, A381775.
Cf. A000108.

Programs

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

Formula

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

A381775 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^6 ) )^(1/6), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 27, 523, 11871, 294668, 7747698, 212054604, 5978347887, 172421233231, 5063192676597, 150872475295522, 4550458484780442, 138652322209300991, 4261638256558924407, 131973650298641750844, 4113788296015093994719, 128973000885015536107140
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A054727, A060941, A381772, A381773, A381774.
Cf. A000108.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^6)/x)^(1/6))

Formula

G.f. A(x) satisfies A(x) = (1 + x*A(x)^6) * C(x*A(x)^6).
a(n) = Sum_{k=0..n} binomial(6*n+2*k+1,k) * binomial(6*n+1,n-k)/(6*n+2*k+1).
a(n) = binomial(1 + 6*n, n)*hypergeom([-n, 1/2+3*n, 1+3*n], [2+5*n, 2+6*n], -4)/(1 + 6*n). - Stefano Spezia, Mar 07 2025

A381782 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * C(x), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 9, 52, 342, 2437, 18331, 143320, 1153308, 9489487, 79470647, 675149665, 5804359859, 50402807459, 441433999816, 3894774605660, 34585663823538, 308867647484634, 2772256164853972, 24994569816424301, 226261997160303326, 2055711320495566962
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A212071, A381773, A381783.
Cf. A000108.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(3*n-k+1,k) * binomial(3*n-3*k+1,n-k)/(3*n-k+1).

A381783 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * C(x*A(x)), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 11, 79, 645, 5682, 52643, 505575, 4987933, 50250625, 514787110, 5346336739, 56161123273, 595667090038, 6370314162095, 68616488830785, 743733580011957, 8106009997644507, 88783190884441892, 976705067814061730, 10787334777299825522, 119569153425125828365
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A212071, A381773, A381782.
Cf. A000108.

Programs

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

Formula

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

A381786 G.f. A(x) satisfies A(x) = (1 + x) * C(x*A(x)^3), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 9, 76, 744, 7986, 90836, 1075714, 13122656, 163769229, 2080985186, 26832199993, 350187469872, 4617094718728, 61406081813812, 822834184073768, 11098254270705028, 150555545320009712, 2052839917410937693, 28118478688846531072, 386727880988105218913, 5338557108832658927346
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A234461, A381773, A381779, A381780.
Cf. A000108.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(3*k+1, n-k)/(5*k+1));

Formula

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

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).
Showing 1-7 of 7 results.

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