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 1-9 of 9 results.

A366400 G.f. A(x) satisfies A(x) = (1 + x * A(x)^(5/2)) / (1 - x).

Original entry on oeis.org

1, 2, 7, 32, 167, 942, 5593, 34438, 217888, 1407938, 9252168, 61641846, 415412036, 2826736736, 19395080061, 134034296976, 932110471089, 6518146460274, 45805553781349, 323313555424924, 2291130483593189, 16294149468133930, 116259325138469680
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A349311, A366401, A366402, A366403, A366404, A366405, A366406, A366407.
Cf. A262441, A370474.
Cf. A219537, A378890.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k/2, n-k)*binomial(5*k/2, k)/(3*k/2+1));
  • PARI
    a(n, r=2, s=-1, t=4, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 12 2024

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k/2,n-k) * binomial(5*k/2,k) / (3*k/2+1).
From Seiichi Manyama, Dec 12 2024: (Start)
G.f. A(x) satisfies:
(1) A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^2.
(2) A(x) = 1/( 1 - x*A(x)^(3/2)/(1 + x*A(x)) )^2.
(3) A(x) = 1 + x * A(x) * (1 + A(x)^(3/2)).
(4) A(x) = B(x)^2 where B(x) is the g.f. of A219537.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)
G.f.: Sum_{k>=0} binomial(5*k/2, k)*x^k/((3*k/2 + 1)*(1 - x)^(5*k/2 + 1)). - Miles Wilson, Feb 02 2025

A366401 G.f. A(x) satisfies A(x) = (1 + x * A(x)^(7/2)) / (1 - x).

Original entry on oeis.org

1, 2, 9, 58, 436, 3572, 30935, 278532, 2581043, 24453404, 235790159, 2306367444, 22829030276, 228240387070, 2301498462245, 23379656027868, 239038022347243, 2457891085704180, 25400777844198274, 263685720722690420, 2748421883496133866
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A349313, A366400, A366402, A366403, A366404, A366405, A366406, A366407.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k/2, n-k)*binomial(7*k/2, k)/(5*k/2+1));

Formula

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

A366402 G.f. A(x) satisfies A(x) = (1 + x * A(x)^(9/2)) / (1 - x).

Original entry on oeis.org

1, 2, 11, 92, 905, 9734, 110867, 1314140, 16041947, 200302394, 2546194497, 32840654064, 428708791851, 5653487876454, 75201937732737, 1007829909427734, 13594917784717860, 184440900147250722, 2515052824018153080, 34451608720123170686, 473853214173320181668
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A366400, A366401, A366403, A366404, A366405, A366406, A366407.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+7*k/2, n-k)*binomial(9*k/2, k)/(7*k/2+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+7*k/2,n-k) * binomial(9*k/2,k) / (7*k/2+1).

A366405 G.f. A(x) satisfies A(x) = (1 + x / A(x)^(5/2)) / (1 - x).

Original entry on oeis.org

1, 2, -3, 22, -138, 1012, -7839, 63506, -531024, 4549276, -39723484, 352237844, -3163252976, 28711196184, -262964888021, 2427319896584, -22557930343459, 210889624536396, -1981972609174109, 18714482015314016, -177453862702083994, 1689045253793239952
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A366400, A366401, A366402, A366403, A366404, A366406, A366407.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(7*k/2-1, k)*binomial(5*k/2-1, n-k)/(7*k/2-1));

Formula

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

A366406 G.f. A(x) satisfies A(x) = (1 + x / A(x)^(7/2)) / (1 - x).

Original entry on oeis.org

1, 2, -5, 44, -383, 3782, -39653, 434324, -4910009, 56862170, -671131131, 8043570088, -97629201137, 1197607836678, -14824033357867, 184923041147906, -2322472423266102, 29341825623660226, -372652945642370654, 4755048678561786946, -60929667733382420198
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A366400, A366401, A366402, A366403, A366404, A366405, A366407.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(9*k/2-1, k)*binomial(7*k/2-1, n-k)/(9*k/2-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(9*k/2-1,k) * binomial(7*k/2-1,n-k) / (9*k/2-1).

A366403 G.f. A(x) satisfies A(x) = (1 + x / sqrt(A(x))) / (1 - x).

Original entry on oeis.org

1, 2, 1, 2, 0, 4, -5, 16, -35, 92, -231, 604, -1584, 4214, -11297, 30538, -83096, 227476, -625991, 1730788, -4805594, 13393690, -37458329, 105089230, -295673993, 834086422, -2358641375, 6684761126, -18985057350, 54022715452, -154000562757, 439742222072
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A366363, A366400, A366401, A366402, A366404, A366405, A366406, A366407.

Programs

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

Formula

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

A366407 G.f. A(x) satisfies A(x) = (1 + x / A(x)^(9/2)) / (1 - x).

Original entry on oeis.org

1, 2, -7, 74, -820, 10196, -134785, 1860668, -26508457, 386843804, -5753126477, 86878155652, -1328593620692, 20533664196478, -320220157730975, 5032648114664896, -79629405527982623, 1267425784159379572, -20279086501234998596, 325989622456860054852
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A366400, A366401, A366402, A366403, A366404, A366405, A366406.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(11*k/2-1, k)*binomial(9*k/2-1, n-k)/(11*k/2-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(11*k/2-1,k) * binomial(9*k/2-1,n-k) / (11*k/2-1).

A366433 G.f. A(x) satisfies A(x) = 1 + x * ((1 - x) / A(x))^(3/2).

Original entry on oeis.org

1, 1, -3, 9, -37, 171, -849, 4421, -23820, 131676, -742616, 4255944, -24714276, 145103426, -859920585, 5137093695, -30902681230, 187034086170, -1138106903928, 6958662440416, -42729903714420, 263400623938140, -1629378251621535, 10111374706286895
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366404.
Cf. A366431, A366432, A366434, A366435, A366436, A366437.

Programs

  • Mathematica
    Table[(-1)^(n-1) * Sum[Binomial[5*k/2 - 1, k]*Binomial[3*k/2, n - k]/(5*k/2 - 1), {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 10 2023 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k/2-1, k)*binomial(3*k/2, n-k)/(5*k/2-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k/2-1,k) * binomial(3*k/2,n-k) / (5*k/2-1).
a(n) ~ -(-1)^n * sqrt(4*10^(1/3) + 10^(2/3) - 5) * 3^(n + 1/2) * 5^(n-1) / (sqrt(Pi) * (2 + 10^(1/3)) * n^(3/2) * (4*10^(1/3) + 10^(2/3) - 11)^n). - Vaclav Kotesovec, Oct 10 2023

A366452 G.f. A(x) satisfies A(x) = 1 + x + x*A(x)^(5/2).

Original entry on oeis.org

1, 2, 5, 20, 90, 440, 2266, 12110, 66525, 373320, 2130865, 12332512, 72202860, 426861830, 2544727475, 15280236800, 92333523153, 561054410200, 3426075429740, 21013974400920, 129403499560500, 799733464576880, 4958649842375975, 30837325310579350
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2023

Keywords

Crossrefs

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366453, A366454, A366455, A366456.
Cf. A259757, A366404.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*k/2+1, n-k)*binomial(5*k/2, k)/(3*k/2+1));

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366404.
a(n) = Sum_{k=0..n} binomial(3*k/2+1,n-k) * binomial(5*k/2,k) / (3*k/2+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A259757. - Seiichi Manyama, Apr 04 2024
Showing 1-9 of 9 results.

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