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.

A365843 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^3 ).

Original entry on oeis.org

1, 6, 54, 578, 6810, 85278, 1113854, 15004746, 206955378, 2908113974, 41484917958, 599202514578, 8745727050762, 128790559374030, 1911191826600462, 28551332345784730, 429040549473424866, 6480799118506040934, 98349636147075506006, 1498732955394826784226
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Column k=3 of A378238.
Cf. A007297, A365842, A365844, A365845.
Cf. A032349, A365622, A365847.
Cf. A144097.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*(n+1),n-k).
G.f.: B^3, where B is the g.f. of A144097.
a(n) ~ sqrt(8060 + 2651*sqrt(10)) * (223 + 70*sqrt(10))^n / (2 * sqrt(5*Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Nov 28 2024

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

A378237 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+3*r+k,n)/(n+3*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 10, 0, 1, 6, 24, 74, 0, 1, 8, 42, 188, 642, 0, 1, 10, 64, 350, 1680, 6082, 0, 1, 12, 90, 568, 3234, 16212, 60970, 0, 1, 14, 120, 850, 5440, 31878, 164584, 635818, 0, 1, 16, 154, 1204, 8450, 54888, 328426, 1732172, 6826690, 0, 1, 18, 192, 1638, 12432, 87402, 574848, 3494142, 18728352, 74958914, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
   1,     1,      1,      1,      1,      1,       1, ...
   0,     2,      4,      6,      8,     10,      12, ...
   0,    10,     24,     42,     64,     90,     120, ...
   0,    74,    188,    350,    568,    850,    1204, ...
   0,   642,   1680,   3234,   5440,   8450,   12432, ...
   0,  6082,  16212,  31878,  54888,  87402,  131964, ...
   0, 60970, 164584, 328426, 574848, 931770, 1433544, ...

Crossrefs

Columns k=0..1 give A000007, A349310.
Cf. A033877, A071949, A378236, A378238, A378239, A378240.

Programs

  • PARI
    T(n, k, t=1, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A349310.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+3) for n > 0.

A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2*n+2*r+k,n)/(2*n+2*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 12, 0, 1, 6, 28, 100, 0, 1, 8, 48, 248, 968, 0, 1, 10, 72, 452, 2480, 10208, 0, 1, 12, 100, 720, 4680, 26688, 113792, 0, 1, 14, 132, 1060, 7728, 51504, 301648, 1318832, 0, 1, 16, 168, 1480, 11800, 87104, 591312, 3531424, 15732064, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024 based on suggestions from Mikhail Kurkov

Keywords

Examples

			Square array begins:
  1,      1,      1,      1,       1,       1,       1, ...
  0,      2,      4,      6,       8,      10,      12, ...
  0,     12,     28,     48,      72,     100,     132, ...
  0,    100,    248,    452,     720,    1060,    1480, ...
  0,    968,   2480,   4680,    7728,   11800,   17088, ...
  0,  10208,  26688,  51504,   87104,  136352,  202560, ...
  0, 113792, 301648, 591312, 1017184, 1621280, 2454256, ...

Crossrefs

Columns k=0..4 give A000007, A219534, A371693, A378155, A378156.
Cf. A033877, A071949, A378236, A378237, A378238, A378240.

Programs

  • PARI
    T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A219534.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-1,k+3) for n > 0.

A378236 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(n+2*r+k,n)/(n+2*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 20, 44, 0, 1, 8, 36, 120, 280, 0, 1, 10, 56, 236, 800, 1936, 0, 1, 12, 80, 400, 1656, 5696, 14128, 0, 1, 14, 108, 620, 2960, 12192, 42416, 107088, 0, 1, 16, 140, 904, 4840, 22592, 92960, 326304, 834912, 0, 1, 18, 176, 1260, 7440, 38352, 176800, 727824, 2572992, 6652608, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
   1,     1,     1,     1,      1,      1,      1, ...
   0,     2,     4,     6,      8,     10,     12, ...
   0,     8,    20,    36,     56,     80,    108, ...
   0,    44,   120,   236,    400,    620,    904, ...
   0,   280,   800,  1656,   2960,   4840,   7440, ...
   0,  1936,  5696, 12192,  22592,  38352,  61248, ...
   0, 14128, 42416, 92960, 176800, 308560, 507152, ...

Crossrefs

Columns k=0..1 give A000007, A346626.
Cf. A033877, A071949, A378237, A378238, A378239, A378240.

Programs

  • PARI
    T(n, k, t=1, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A346626.
B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+2) for n > 0.

A378240 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*n+3*r+k,n)/(3*n+3*r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 18, 0, 1, 6, 40, 234, 0, 1, 8, 66, 540, 3570, 0, 1, 10, 96, 926, 8400, 59586, 0, 1, 12, 130, 1400, 14706, 141876, 1053570, 0, 1, 14, 168, 1970, 22720, 251622, 2528760, 19392490, 0, 1, 16, 210, 2644, 32690, 394152, 4524786, 46815116, 367677090, 0
Offset: 0

Views

Author

Seiichi Manyama, Nov 20 2024

Keywords

Examples

			Square array begins:
  1,       1,       1,       1,       1,        1,        1, ...
  0,       2,       4,       6,       8,       10,       12, ...
  0,      18,      40,      66,      96,      130,      168, ...
  0,     234,     540,     926,    1400,     1970,     2644, ...
  0,    3570,    8400,   14706,   22720,    32690,    44880, ...
  0,   59586,  141876,  251622,  394152,   575402,   801948, ...
  0, 1053570, 2528760, 4524786, 7156128, 10553970, 14867704, ...

Crossrefs

Columns k=0..1 give A000007, A364167.
Cf. A033877, A071949, A378236, A378237, A378238, A378239.

Programs

  • PARI
    T(n, k, t=3, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(3/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A364167.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+5). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+5) for n > 0.
Showing 1-6 of 6 results.

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