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 1-8 of 8 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).

A366404 G.f. A(x) satisfies A(x) = (1 + x / A(x)^(3/2)) / (1 - x).

Original entry on oeis.org

1, 2, -1, 8, -29, 142, -707, 3714, -20106, 111570, -631046, 3624898, -21089378, 124014048, -735906537, 4401187158, -26501494072, 160532592098, -977574311830, 5981088128586, -36748815585834, 226651808352306, -1402726443269229, 8708648263017666
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A366400, A366401, A366402, A366403, A366405, A366406, A366407.

Programs

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

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).

A366437 G.f. A(x) satisfies A(x) = 1 + x * ((1 - x) / A(x))^(7/2).

Original entry on oeis.org

1, 1, -7, 49, -427, 4165, -43435, 473977, -5344333, 61772179, -727993301, 8714701219, -105672771225, 1295237037815, -16021641194545, 199747074505773, -2507395464414008, 31664298046926328, -401994771266030880, 5127701624204157600, -65684716411944207144
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Partial sums give A366406.
Cf. A366431, A366432, A366433, A366434, A366435, A366436.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(9*k/2-1, k)*binomial(7*k/2, 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,n-k) / (9*k/2-1).
Showing 1-8 of 8 results.

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