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.

A381773 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, 15, 157, 1913, 25427, 357546, 5229980, 78765793, 1213181593, 19021747383, 302595975502, 4871780511910, 79232327379407, 1299767617080662, 21481625997258747, 357350097625089497, 5978708468143961925, 100537111802285439375, 1698302173359384479307
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A054727, A060941, A381772, A381774, A381775.
Cf. A212071, A381782, A381783.
Cf. A234461, A381779, A381780, A381786.
Cf. A000108.

Programs

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

Formula

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

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).

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

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).

A381784 G.f. A(x) satisfies A(x) = (1 + x*A(x)^4) * C(x*A(x)^2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 15, 153, 1799, 22969, 309479, 4331175, 62349575, 917335467, 13732751589, 208509835114, 3203279694575, 49701110565986, 777708690091907, 12258870836704797, 194475105262057575, 3102607480658510165, 49746656826517452788, 801205735002960886531, 12956005807148939155717
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A234461, A381774.
Cf. A000108.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(4*n-2*k+1, n-k))/(4*n+1);

Formula

a(n) = (1/(4*n+1)) * Sum_{k=0..n} binomial(4*n+1,k) * binomial(4*n-2*k+1,n-k).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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