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.

A364748 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)).

Original entry on oeis.org

1, 1, 6, 47, 424, 4159, 43097, 464197, 5145475, 58313310, 672598269, 7869856070, 93183973405, 1114471042413, 13443614108307, 163372291277764, 1998239045199623, 24580340878055298, 303893356012560280, 3774099648814193998, 47061518776483143441
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A000108, A001003, A108447, A364747, A378691, A378692.
Cf. A002295, A243667, A365192, A365193, A365194.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(5*n-4*k, n-1-k))/n);
  • PARI
    a(n, r=1, s=1, t=5, u=1) = 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)); \\ Seiichi Manyama, Dec 05 2024

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(5*n-4*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 05 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^4/(1 - x*A(x))).
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). (End)

A364747 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)).

Original entry on oeis.org

1, 1, 5, 32, 234, 1854, 15490, 134380, 1198944, 10931761, 101412677, 954155059, 9083120975, 87326765375, 846709605539, 8269910074087, 81291388929027, 803592049667495, 7983612883739843, 79671910265120574, 798283229227457304, 8027625597750959053
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A000108, A001003, A108447, A364748, A378691, A378692.
Cf. A002294, A243659, A364765, A364792.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);
  • PARI
    a(n, r=1, s=1, t=4, u=1) = 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)); \\ Seiichi Manyama, Dec 05 2024

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 05 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 - x*A(x))).
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). (End)

A378686 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(7/3)/(1 - x*A(x)) )^3.

Original entry on oeis.org

1, 3, 27, 313, 4122, 58584, 875897, 13577139, 216224616, 3516601243, 58160887857, 975211608399, 16539799297342, 283243124783136, 4890858070498203, 85060240453556192, 1488653675438168001, 26197808077514204832, 463311206395709908936, 8229849868810254813378
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A108447, A371581.
Cf. A349017, A369012.
Cf. A378690, A378692, A378693.
Cf. A378685.

Programs

  • PARI
    a(n, r=3, s=1, t=7, 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) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)) )^3.
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A378685.
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).

A378690 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(7/2)/(1 - x*A(x)) )^2.

Original entry on oeis.org

1, 2, 17, 190, 2438, 33938, 498413, 7602010, 119261202, 1912171310, 31194947785, 516153663072, 8641160417191, 146105874059670, 2491396820758004, 42795782630083868, 739842609794223330, 12862556429464405500, 224744883747568868574, 3944534317072930309360
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A378686, A378692, A378693.
Cf. A378688.

Programs

  • PARI
    a(n, r=2, s=1, t=7, u=2) = 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) satisfies A(x) = 1/( 1 - x*A(x)^3/(1 - x*A(x)) )^2.
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A378688.
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).

A378691 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6/(1 - x*A(x)).

Original entry on oeis.org

1, 1, 7, 65, 699, 8192, 101538, 1309007, 17373825, 235820907, 3258327727, 45676003435, 648019627185, 9286982935406, 134247731827970, 1955128344950960, 28659409029300490, 422517375650417841, 6260750426764454787, 93191618760715641120, 1392823412892172416996
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A000108, A001003, A108447, A364747, A364748, A378692.

Programs

  • PARI
    a(n, r=1, s=1, t=6, u=1) = 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) satisfies A(x) = 1/(1 - x*A(x)^5/(1 - x*A(x))).
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).

A378693 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(7/6)/(1 - x*A(x)) )^6.

Original entry on oeis.org

1, 6, 63, 806, 11445, 173388, 2745470, 44891118, 752141682, 12845874594, 222813745704, 3914269052736, 69501455945987, 1245309605501088, 22488056019050124, 408861223600687710, 7478056231521533658, 137496627558561863460, 2540015518588821201453
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A378686, A378690, A378692.
Cf. A378694.

Programs

  • PARI
    a(n, r=6, s=1, t=7, u=6) = 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) satisfies A(x) = 1/( 1 - x*A(x)/(1 - x*A(x)) )^6.
G.f.: A(x) = B(x)^6 where B(x) is the g.f. of A378694.
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).

A378694 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)^6).

Original entry on oeis.org

1, 1, 8, 91, 1210, 17577, 270314, 4326070, 71300386, 1202012254, 20630488004, 359279348424, 6332747550808, 112761701957119, 2025325557546780, 36650364776763804, 667570101840389826, 12229542931765845994, 225183117821591853440, 4165207037639796288385
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2024

Keywords

Crossrefs

Cf. A378685, A378688, A378692.

Programs

  • PARI
    a(n, r=1, s=1, t=7, u=6) = 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) satisfies A(x) = 1/(1 - x*A(x)^6/(1 - x*A(x)^6)).
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 1-7 of 7 results.

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

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