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 1-6 of 6 results.

A365118 G.f. satisfies A(x) = (1 + x / (1 - x*A(x)))^2.

Original entry on oeis.org

1, 2, 3, 8, 23, 72, 237, 808, 2830, 10118, 36779, 135510, 504935, 1899494, 7204238, 27517766, 105761937, 408715018, 1587169591, 6190357852, 24238696551, 95244997612, 375469654543, 1484519159122, 5885302251250, 23389997790804, 93172394487012
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2023

Keywords

Crossrefs

Cf. A001006, A365119.
Cf. A161634, A378801.

Programs

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

Formula

If g.f. satisfies A(x) = (1 + x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*(n-k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
G.f.: A(x) = (1 + x*B(x))^2 where B(x) is the g.f. of A161634. - Seiichi Manyama, Dec 09 2024

A365121 G.f. A(x) satisfies A(x) = (1 + x / (1 - x*A(x))^2)^3.

Original entry on oeis.org

1, 3, 9, 40, 192, 993, 5375, 30081, 172650, 1010640, 6010530, 36214656, 220590082, 1356131892, 8403647454, 52436122717, 329170499604, 2077465903503, 13173914483799, 83897445169341, 536355204428412, 3440875097256529, 22144300030907667
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2023

Keywords

Crossrefs

Cf. A000108, A365120.
Cf. A365119, A365122.
Cf. A367242.

Programs

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

Formula

If g.f. satisfies A(x) = (1 + x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*(n-k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A367242. - Seiichi Manyama, Dec 06 2024

A365122 G.f. A(x) satisfies A(x) = (1 + x / (1 - x*A(x))^3)^3.

Original entry on oeis.org

1, 3, 12, 64, 372, 2319, 15105, 101649, 701073, 4929657, 35207220, 254690517, 1862325262, 13742311074, 102204992352, 765328009950, 5765316776550, 43661497944861, 332217854059362, 2538540859615095, 19471592691620310, 149871698475060433, 1157188723053901449
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2023

Keywords

Crossrefs

Cf. A006013, A365113.
Cf. A365119, A365121.
Cf. A371616.

Programs

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

Formula

If g.f. satisfies A(x) = (1 + x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*(n-k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A371616. - Seiichi Manyama, Dec 06 2024

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).

A378732 G.f. A(x) satisfies A(x) = ( 1 + x / (1 - x*A(x)) )^4.

Original entry on oeis.org

1, 4, 10, 36, 155, 704, 3384, 16844, 86097, 449344, 2384170, 12822556, 69743953, 382982940, 2120323014, 11822279232, 66327376437, 374162700460, 2120999728610, 12075668658000, 69021358842795, 395909382981572, 2278286453089574, 13149207655326372, 76096242994616990
Offset: 0

Views

Author

Seiichi Manyama, Dec 05 2024

Keywords

Crossrefs

Cf. A364743, A371612, A378731.
cf. A365118, A365119.

Programs

  • PARI
    a(n, r=4, s=1, t=0, u=4) = 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 A364743.
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).
Showing 1-6 of 6 results.

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

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