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.

A371486 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 30, 260, 2465, 24796, 260008, 2811216, 31117240, 350890260, 4016744586, 46556054072, 545273713228, 6443442857024, 76727957438650, 919796418086076, 11091249210406816, 134439965189940176, 1637160457090585016, 20019920157735604796, 245733987135102838131
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2024

Keywords

Crossrefs

Cf. A002212, A270386, A371483.
Cf. A118971, A349332.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+3,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^4 ).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349332.

A371517 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 26, 188, 1459, 11892, 100444, 871528, 7722557, 69590628, 635807180, 5876094308, 54836925779, 516029817620, 4891147100886, 46653935716492, 447490869463145, 4313492172957396, 41763413498670702, 405968522259130636, 3960526930400038404
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A045868, A371516, A371520, A371521.
Cf. A349331, A371483, A371518.
Cf. A371486.

Programs

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

Formula

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

A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 15, 82, 477, 2901, 18235, 117555, 773085, 5166478, 34987170, 239570655, 1655933060, 11538839130, 80971109712, 571702698185, 4058556404958, 28951715755830, 207424064434502, 1491898838023884, 10768487956456506, 77977009814421534, 566310026687320290
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2024

Keywords

Crossrefs

Cf. A045868, A371517, A371520, A371521.
Cf. A270386, A307678.
Cf. A371483.

Programs

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

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+2,k)/(2*k+3) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(3*k+3,k)/(k+1).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A307678.

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

Cf. A349333, A371379, A371521, A371522, A371523.
Cf. A270386, A371483, A371486.
Cf. A130564.

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.

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

Cf. A349331, A371483, A371517.
Cf. A270386, A371523.

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.

A382885 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x) * A(x) )^3.

Original entry on oeis.org

1, 3, 18, 121, 900, 7110, 58598, 498153, 4336533, 38463732, 346368351, 3158325168, 29102914959, 270582713670, 2535191045652, 23913087584045, 226892934532149, 2164080724942155, 20737076963936828, 199542537271568802, 1927347504059464995, 18679645863925666721
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2025

Keywords

Crossrefs

Cf. A052709, A371576.
Cf. A365178, A371483.

Programs

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

Formula

G.f. A(x) satisfies A(x) = ( 1 + x * (1+x) * A(x)^(4/3) )^3.
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).
G.f.: B(x)^3, where B(x) is the g.f. of A365178.
Showing 1-6 of 6 results.

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