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 1-3 of 3 results.

A382824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^2 ).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 8, 8, 6, 24, 28, 34, 28, 24, 120, 124, 150, 150, 124, 120, 720, 668, 768, 854, 768, 668, 720, 5040, 4248, 4584, 5204, 5204, 4584, 4248, 5040, 40320, 31176, 31512, 35188, 37556, 35188, 31512, 31176, 40320, 362880, 259488, 246072, 265896, 290380, 290380, 265896, 246072, 259488, 362880
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2025

Keywords

Examples

			Square array begins:
    1,   1,    2,     6,     24,     120, ...
    1,   3,    8,    28,    124,     668, ...
    2,   8,   34,   150,    768,    4584, ...
    6,  28,  150,   854,   5204,   35188, ...
   24, 124,  768,  5204,  37556,  290380, ...
  120, 668, 4584, 35188, 290380, 2546852, ...

Crossrefs

Main diagonal gives A382827.
Cf. A382823, A382825.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*abs(stirling(n+1, j+1, 1)*stirling(k+1, j+1, 1)));

Formula

E.g.f.: 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^2 ).
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} j! * (j+1)! * |Stirling1(n+1,j+1)| * |Stirling1(k+1,j+1)|.

A382825 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^3 ).

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 11, 11, 6, 24, 39, 55, 39, 24, 120, 174, 255, 255, 174, 120, 720, 942, 1338, 1623, 1338, 942, 720, 5040, 6012, 8106, 10434, 10434, 8106, 6012, 5040, 40320, 44244, 56292, 72762, 82116, 72762, 56292, 44244, 40320, 362880, 369072, 442860, 560988, 668580, 668580, 560988, 442860, 369072, 362880
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2025

Keywords

Examples

			Square array begins:
    1,   1,    2,     6,     24,     120, ...
    1,   4,   11,    39,    174,     942, ...
    2,  11,   55,   255,   1338,    8106, ...
    6,  39,  255,  1623,  10434,   72762, ...
   24, 174, 1338, 10434,  82116,  668580, ...
  120, 942, 8106, 72762, 668580, 6302028, ...

Crossrefs

Main diagonal gives A382828.
Cf. A382823, A382824.
Cf. A382673, A382800.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*abs(stirling(n+1, j+1, 1)*stirling(k+1, j+1, 1)));

Formula

E.g.f.: 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^3 ).
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * |Stirling1(n+1,j+1)| * |Stirling1(k+1,j+1)|.

A382826 a(n) = Sum_{k=0..n} (k! * Stirling1(n+1,k+1))^2.

Original entry on oeis.org

1, 2, 17, 337, 12152, 696076, 58136500, 6673107316, 1008077743552, 193915431216576, 46281189562936704, 13420575661095930240, 4647502230640182602496, 1894412230202331489632256, 897850527136410029486517504, 489578762044356075253626875136
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2025

Keywords

Crossrefs

Main diagonal of A382823.
Cf. A048163, A382792.

Programs

  • PARI
    a(n) = sum(k=0, n, (k!*stirling(n+1, k+1, 1))^2);

Formula

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y)) ).
a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1+x) * (1+y) * (1 - log(1+x) * log(1+y)) ).
Showing 1-3 of 3 results.

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