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 1-5 of 5 results.

A362079 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^n)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 10, 0, 1, 4, 12, 28, 45, 0, 1, 5, 18, 55, 145, 251, 0, 1, 6, 25, 92, 315, 896, 1624, 0, 1, 7, 33, 140, 571, 2106, 6328, 11908, 0, 1, 8, 42, 200, 930, 4076, 15946, 50212, 97545, 0, 1, 9, 52, 273, 1410, 7026, 32718, 134730, 441489, 880660, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Examples

			Square array begins:
  1,   1,   1,    1,    1,    1, ...
  0,   1,   2,    3,    4,    5, ...
  0,   3,   7,   12,   18,   25, ...
  0,  10,  28,   55,   92,  140, ...
  0,  45, 145,  315,  571,  930, ...
  0, 251, 896, 2106, 4076, 7026, ...

Crossrefs

Columns k=0..3 give A000007, A099237, A362084, A362085.
Main diagonal gives A362080.
Cf. A362078.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(n*j, n-j));

Formula

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(n*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(n*j,n-j).

A362080 a(n) = [x^n] 1/(1 - x*(1+x)^n)^n.

Original entry on oeis.org

1, 1, 7, 55, 571, 7026, 98925, 1562219, 27214867, 516646333, 10589130223, 232574622116, 5440521381816, 134859301929873, 3527034780915985, 96965997588549555, 2793286163779275779, 84076751617833902070, 2637677096916448507104
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Links

Crossrefs

Main diagonal of A362078.
Main diagonal of A362079.
Cf. A099237.

Programs

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

Formula

a(n) = Sum_{k=0..n} (-1)^k * binomial(-n,k) * binomial(n*k,n-k) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n*k,n-k).

A362087 a(n) = Sum_{k=0..n} (-1)^k * binomial(-n,k) * binomial(2*k,n-k).

Original entry on oeis.org

1, 1, 7, 37, 215, 1271, 7651, 46614, 286599, 1774630, 11050897, 69134572, 434174819, 2735565574, 17283825370, 109466361512, 694764983463, 4417771590123, 28137563496298, 179478199605550, 1146342590242465, 7330598365285470, 46928753892901140
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Crossrefs

Column k=2 of A362078.
Cf. A362084.

Programs

  • Mathematica
    Table[Sum[Binomial[n + k - 1, k]*Binomial[2*k, n-k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 08 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(2*k, n-k));

Formula

a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(2*k,n-k).
a(n) = [x^n] 1/(1 - x*(1+x)^2)^n.

A362088 a(n) = Sum_{k=0..n} (-1)^k * binomial(-n,k) * binomial(3*k,n-k).

Original entry on oeis.org

1, 1, 9, 55, 369, 2526, 17577, 123810, 879921, 6297643, 45325764, 327720471, 2378598885, 17319785588, 126463925310, 925620561780, 6789081127137, 49887667836693, 367189338574827, 2706613194698783, 19977334121704344, 147628150691297100
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Crossrefs

Column k=3 of A362078.
Cf. A362085.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(3*k,n-k).
a(n) = [x^n] 1/(1 - x*(1+x)^3)^n.

A362125 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 47, 8, 0, 1, 6, 40, 124, 198, 118, 13, 0, 1, 7, 57, 235, 571, 681, 290, 21, 0, 1, 8, 77, 398, 1320, 2500, 2263, 702, 34, 0, 1, 9, 100, 623, 2640, 7026, 10504, 7341, 1677, 55, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2023

Keywords

Examples

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   7,  15,   26,   40, ...
  0, 3,  18,  55,  124,  235, ...
  0, 5,  47, 198,  571, 1320, ...
  0, 8, 118, 681, 2500, 7026, ...

Crossrefs

Columns k=0..3 give A000007, A000045(n+1), A362126, A382614.
Main diagonal gives A362080.
Cf. A362078, A362079.

Programs

  • PARI
    T(n, k) = sum(j=0, n, binomial(j+k-1, j)*binomial(k*j, n-j));

Formula

T(n,k) = Sum_{j=0..n} (-1)^j * binomial(-k,j) * binomial(k*j,n-j) = Sum_{j=0..n} binomial(j+k-1,j) * binomial(k*j,n-j).
Showing 1-5 of 5 results.

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