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.

A371521 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^6.

Original entry on oeis.org

1, 6, 57, 614, 7158, 88002, 1123689, 14760024, 198172050, 2707560544, 37522666803, 526190125308, 7452866846847, 106465245105972, 1532129408941797, 22191180837313808, 323243244688652943, 4732225866305323686, 69591395772704207770, 1027547992261749954798
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 6*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+5, k)/(5*k+6));

Formula

a(n) = 6 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+5,k)/(5*k+6) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+6,k)/(k+1).
G.f.: A(x) = B(x)^6 where B(x) is the g.f. of A349333.

A371379 G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/2) / (1-x))^4.

Original entry on oeis.org

1, 4, 34, 344, 3859, 46240, 579722, 7511272, 99782617, 1351784792, 18604380884, 259395030992, 3656180724752, 52011780756632, 745799171500502, 10768038899631476, 156414710103922340, 2284233700081510820, 33517461646190624690, 493917761019513208800
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+3, k)/(5*k+4));

Formula

a(n) = 4 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+3,k)/(5*k+4).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349333.

A371519 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 45, 470, 5375, 65231, 825225, 10764185, 143739440, 1955340360, 27001732972, 377530388235, 5333865386885, 76031188364860, 1092117166466660, 15792298241897649, 229704197116753825, 3358528175751886765, 49333470827844265285, 727680248026484478405
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+4, k)/(k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+4,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^5 ).
G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349333.

A371522 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1-x))^3.

Original entry on oeis.org

1, 3, 24, 235, 2586, 30603, 380359, 4896753, 64731747, 873539236, 11984536632, 166661420814, 2343950447112, 33282048811530, 476462982915993, 6869620848003570, 99663539644072305, 1453861111238442363, 21312207036239313936, 313783619269186619589
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 3*sum(k=0, n, binomial(n-1, n-k)*binomial(6*k+2, k)/(5*k+3));

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(6*k+2,k)/(5*k+3).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349333.

A371518 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2 / (1-x))^2.

Original entry on oeis.org

1, 2, 11, 72, 525, 4104, 33647, 285526, 2486809, 22103726, 199697284, 1828472914, 16929944932, 158246198836, 1491210732346, 14151603542612, 135130396860130, 1297381593071890, 12516650939119421, 121281286192026308, 1179769340479567499
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = 2 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+1,k)/(3*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349331.

A371537 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3 / (1+x))^2.

Original entry on oeis.org

1, 2, 11, 90, 845, 8620, 92792, 1037474, 11930952, 140223730, 1676824810, 20336742860, 249554057158, 3092735367966, 38653949888993, 486656046354650, 6166315484899445, 78573243500307870, 1006223574171080479, 12943581721362983708, 167170200918998754129
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (-1)^(n-k)*binomial(n-1, n-k)*binomial(6*k+2, k)/(6*k+2));

Formula

a(n) = 2 * Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,n-k) * binomial(6*k+2,k)/(6*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349362.

A371583 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) / (1 - x) )^2.

Original entry on oeis.org

1, 2, 13, 104, 940, 9166, 94044, 1000602, 10939780, 122161128, 1387361151, 15974899766, 186069556707, 2188416960148, 25953579753464, 310022550197360, 3726709235290628, 45047517497268968, 547217895030263028, 6676784544374859088, 81789906534091716353
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=2, s=1, t=5, u=0) = 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) = 2 * Sum_{k=0..n} binomial(5*k+2,k) * binomial(n-1,n-k)/(5*k+2).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A349332.
Showing 1-7 of 7 results.