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 1-6 of 6 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).

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).

A381818 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, 12, 97, 903, 9129, 97419, 1080058, 12319200, 143630575, 1704099034, 20507897766, 249734145622, 3071587654688, 38102046141882, 476138815310364, 5988435287060671, 75745116484532586, 962898676577135634, 12295850972794555196, 157649023155654522723, 2028662477759375282902
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A381817, A381819, A381820.
Cf. A364592, A381830, A381831.
Cf. A000108, A381772.

Programs

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

Formula

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

A381780 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * C(x*A(x)^3), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 13, 122, 1348, 16317, 209366, 2797461, 38509302, 542367569, 7778173646, 113196865436, 1667497600735, 24816081138489, 372551391235504, 5635157636123317, 85797446797707896, 1313857342649814042, 20222887980813290849, 312694810135597988049, 4854881337618505385339
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A007852, A069271, A381772.
Cf. A000108.

Programs

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

Formula

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

A381785 G.f. A(x) satisfies A(x) = (1 + x) * C(x*A(x)^2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 7, 45, 335, 2731, 23573, 211741, 1958571, 18529392, 178459000, 1743868792, 17246702932, 172302244669, 1736302280083, 17627794322287, 180133941044517, 1851310247393202, 19123511540724822, 198437973436950204, 2067524004169000212, 21620908821378509071
Offset: 0

Views

Author

Seiichi Manyama, Mar 07 2025

Keywords

Crossrefs

Cf. A212071, A381772, A381778, A381784.
Cf. A000108.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(2*k+1,n-k)/(4*k+1).
Showing 1-6 of 6 results.

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