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 1-10 of 11 results. Next

A364743 G.f. satisfies A(x) = 1 / (1 - x*(1 + x*A(x))^4).

Original entry on oeis.org

1, 1, 5, 19, 85, 402, 1971, 9976, 51633, 272131, 1455486, 7879664, 43096967, 237777710, 1321792096, 7396125088, 41624735353, 235461758085, 1338049873395, 7634930866465, 43726638130854, 251273386911443, 1448362622788376, 8371936106228253
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A001006, A161634, A364742, A364744.

Programs

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

Formula

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

A364744 G.f. satisfies A(x) = 1 / (1 - x*(1 + x*A(x))^5).

Original entry on oeis.org

1, 1, 6, 26, 131, 706, 3932, 22618, 133099, 797545, 4850296, 29859028, 185712831, 1165227025, 7366475715, 46877977451, 300049605259, 1930395961235, 12476394685445, 80968876247330, 527424073700966, 3447190219684125, 22599794010813360
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A001006, A161634, A364742, A364743.

Programs

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

Formula

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

A364761 G.f. satisfies A(x) = 1 / (1 + x*(1 + x*A(x))^3).

Original entry on oeis.org

1, -1, -2, 5, 12, -41, -89, 391, 733, -4051, -6320, 44120, 54990, -496406, -465932, 5710408, 3637847, -66714699, -22683218, 787957397, 35371351, -9376925921, 2356626520, 112147043475, -61910867756, -1345231820826, 1158452138826, 16156200619772
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Crossrefs

Cf. A007440, A364760, A364762, A364763.
Cf. A364742.

Programs

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

Formula

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

A367242 G.f. satisfies A(x) = 1 + x / (1 - x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 2, 9, 40, 202, 1068, 5884, 33356, 193365, 1140940, 6829601, 41372238, 253156085, 1562416632, 9714660195, 60795387840, 382639224327, 2420498032350, 15380899180204, 98134455984896, 628425763698123, 4037685422823604, 26021345223509038, 168164609160791154
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Crossrefs

Cf. A364742.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).

A371616 G.f. satisfies A(x) = 1 + x / (1 - x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 3, 15, 82, 495, 3147, 20812, 141621, 985287, 6976369, 50108232, 364202415, 2673756449, 19797659586, 147677816532, 1108711280376, 8371222635096, 63525564996093, 484243596619753, 3706268752629237, 28470703720193010, 219432896755734137
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2024

Keywords

Links

Crossrefs

Cf. A364742, A367242.
Cf. A001764, A365113.
Cf. A002293, A367281.

Programs

  • PARI
    a(n, r=1, s=3, t=0, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

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

A365131 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^3)^2.

Original entry on oeis.org

1, 2, 11, 62, 395, 2662, 18720, 135738, 1007607, 7619456, 58488028, 454556544, 3569655975, 28282204680, 225796917864, 1814732935968, 14670580718486, 119215212413412, 973246346463636, 7978384233270126, 65649676250344747, 542031604244083664
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2023

Keywords

Crossrefs

Cf. A364742, A365132.
Cf. A137952.

Programs

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

Formula

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).

A365132 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^3)^3.

Original entry on oeis.org

1, 3, 21, 163, 1410, 12954, 124197, 1228269, 12438504, 128338224, 1344328020, 14258394921, 152820980120, 1652596758738, 18008899150278, 197566103218974, 2180167982738235, 24183969704272350, 269513577777159816, 3016075156973165367, 33879382051847177781
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2023

Keywords

Crossrefs

Cf. A364742, A365131.
Cf. A001764.

Programs

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

Formula

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).

A378730 G.f. A(x) satisfies A(x) = ( 1 + x / (1 - x*A(x)^(3/2)) )^2.

Original entry on oeis.org

1, 2, 3, 10, 35, 134, 544, 2288, 9907, 43830, 197300, 900738, 4160521, 19408084, 91302317, 432663728, 2063421045, 9896113574, 47698770359, 230932635206, 1122545149941, 5476405604806, 26805046064328, 131595640014314, 647829955225386, 3197267300375652
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2024

Keywords

Crossrefs

Cf. A364742, A365119.

Programs

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

Formula

G.f.: A(x) = (1 + x*B(x))^2 where B(x) is the g.f. of A364742.
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(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A378858 G.f. A(x) satisfies A(x) = ( 1 + x/(1 - x*A(x)^(3/4)) )^4.

Original entry on oeis.org

1, 4, 10, 32, 119, 468, 1934, 8256, 36135, 161276, 731158, 3357748, 15587004, 73021200, 344786056, 1639145180, 7839483967, 37692820908, 182087119582, 883358016328, 4301799946048, 21021519618724, 103049029114618, 506608410994868, 2497162797380145, 12338908560964968
Offset: 0

Views

Author

Seiichi Manyama, Dec 09 2024

Keywords

Crossrefs

Cf. A364742, A365119, A378730.
Cf. A005554, A378801.

Programs

  • PARI
    a(n, r=4, s=1, t=0, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f.: A(x) = (1 + x*B(x))^4 where B(x) is the g.f. of A364742.
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(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

A367260 G.f. satisfies A(x) = 1 + x*A(x)^3 * (1 + x*A(x))^3.

Original entry on oeis.org

1, 1, 6, 36, 251, 1881, 14817, 120950, 1014042, 8680377, 75552553, 666614637, 5948817600, 53599239101, 486926148000, 4455202562652, 41018936164660, 379747493741643, 3532914858433284, 33012260400580342, 309692626084981245, 2915659701275923491
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Crossrefs

Cf. A019497, A361305, A364742.
Cf. A367233.

Programs

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

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).
Showing 1-10 of 11 results. Next

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

Content is available under The OEIS End-User License Agreement.