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 1-4 of 4 results.

A371675 G.f. satisfies A(x) = 1 + x * A(x)^(3/2) * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 32, 324, 3696, 45316, 583152, 7769348, 106250144, 1482925956, 21037812352, 302478044996, 4397824031376, 64549296707460, 955150116019920, 14233474784850948, 213417133281087040, 3217460713030341892, 48741781832765496288, 741606216370357708612
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Crossrefs

Column k=2 of A378238.
Cf. A006319, A032349, A371676. A371677, A371678.
Cf. A144097, A371679.

Programs

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

Formula

G.f. satisfies A(x) = ( 1 + x * A(x)^(3/2) * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A144097.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(3*n+k+2,n)/(3*n+k+2).
a(n) ~ sqrt((88 + 161*sqrt(2/5))/Pi) * (223 + 70*sqrt(10))^n / (n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Nov 28 2024

A371676 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 40, 524, 7824, 126228, 2143544, 37750812, 683194912, 12628104740, 237388091208, 4524456276524, 87228274533040, 1698091537435444, 33332913873239640, 659038408936005692, 13112372856351746112, 262338658739430857796, 5274545338183090647656
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Links

Crossrefs

Cf. A006319, A032349, A371675. A371677, A371678.
Cf. A260332.

Programs

  • PARI
    a(n, r=2, t=4, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

G.f. satisfies A(x) = ( 1 + x * A(x)^2 * (1 + A(x)^(1/2)) )^2.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k+2,n)/(4*n+k+2).

A371677 G.f. satisfies A(x) = 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 48, 772, 14256, 285380, 6023552, 131991940, 2974096544, 68475379204, 1603913377040, 38099316926340, 915619571011024, 22222175033464260, 543894269096547296, 13409307961403740420, 332707806061304185408, 8301493488646359256580
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Crossrefs

Cf. A006319, A032349, A371675. A371676, A371678.
Cf. A363006.

Programs

  • PARI
    a(n, r=2, t=5, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

G.f. satisfies A(x) = ( 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A363006.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(5*n+k+2,n)/(5*n+k+2).

A371700 G.f. satisfies A(x) = 1 + x * A(x)^6 * (1 + A(x)).

Original entry on oeis.org

1, 2, 26, 482, 10450, 247554, 6208970, 162064322, 4356511138, 119788611458, 3353361311738, 95251219926690, 2738421518770546, 79531905952256642, 2329955712706784682, 68770993359030211458, 2043143866891345880898, 61050342965542475675906
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2024

Keywords

Crossrefs

Cf. A006318, A027307, A144097, A260332, A363006.
Cf. A371678.

Programs

  • PARI
    a(n, r=1, t=6, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+1,n)/(6*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 2^(n-k) * binomial(n,k) * binomial(7*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(6*n,k-1) for n > 0.
Showing 1-4 of 4 results.

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

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