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.

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).

A364594 G.f. satisfies A(x) = 1/(1-x) + x^2*(1-x)*A(x)^4.

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 98, 316, 1065, 3649, 12775, 45299, 162713, 590097, 2159015, 7957003, 29517141, 110116277, 412879256, 1555048142, 5880591163, 22319380999, 84992915958, 324634976440, 1243396473153, 4774504667881, 18376620653851, 70883537152927
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A110199, A364593.
Cf. A364592, A364596.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n, 2*k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * binomial(4*k,k) / (3*k+1).

A364596 G.f. satisfies A(x) = 1/(1-x) + x^3*(1-x)*A(x)^4.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 15, 36, 82, 191, 471, 1166, 2884, 7267, 18523, 47349, 121821, 315781, 822165, 2148811, 5641035, 14864295, 39287907, 104154066, 276899112, 737984583, 1971375679, 5277570860, 14156881590, 38045460023, 102421374775, 276174537027, 745822179831
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A364595, A364597.
Cf. A215340, A226974.
Cf. A364592, A364594.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-k, 2*k)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-k,2*k) * binomial(4*k,k) / (3*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).

A381943 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^2, where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 11, 60, 425, 3426, 29619, 267738, 2497889, 23866056, 232325475, 2295889266, 22971682893, 232248775669, 2368969672183, 24348849065860, 251930963865061, 2621914660411919, 27428338267887815, 288258167672381602, 3042002859317810001, 32222429872821051817
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Partial sums of A364592.
Cf. A086616, A381867, A381945.
Cf. A001764.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(4*k+1,k) * binomial(n+k+1,n-k)/(4*k+1).

A381832 G.f. A(x) satisfies A(x) = C(x*A(x)^3) / (1 - x), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 10, 81, 796, 8616, 98973, 1184324, 14602486, 184219731, 2366543116, 30851212416, 407106050261, 5427274340091, 72986372975716, 988937692146346, 13487903251385562, 185022817888443780, 2551096865411701371, 35335463473311506321, 491444773227779518956, 6860346682881319595632
Offset: 0

Views

Author

Seiichi Manyama, Mar 08 2025

Keywords

Crossrefs

Cf. A014137, A188687, A364592.
Cf. A000108, A381786.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(n+2*k, n-k)/(5*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(n+2*k,n-k)/(5*k+1).
Showing 1-7 of 7 results.

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