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.

A382735 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (exp(x) + exp(y) - exp(x+y))^3.

Original entry on oeis.org

1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 3, 27, 3, 0, 0, 3, 75, 75, 3, 0, 0, 3, 171, 579, 171, 3, 0, 0, 3, 363, 2667, 2667, 363, 3, 0, 0, 3, 747, 10083, 22779, 10083, 747, 3, 0, 0, 3, 1515, 34635, 142923, 142923, 34635, 1515, 3, 0, 0, 3, 3051, 112899, 761211, 1396803, 761211, 112899, 3051, 3, 0
Offset: 0

Views

Author

Seiichi Manyama, Apr 04 2025

Keywords

Examples

			Square array begins:
  1, 0,   0,     0,      0,       0, ...
  0, 3,   3,     3,      3,       3, ...
  0, 3,  27,    75,    171,     363, ...
  0, 3,  75,   579,   2667,   10083, ...
  0, 3, 171,  2667,  22779,  142923, ...
  0, 3, 363, 10083, 142923, 1396803, ...

Crossrefs

Main diagonal gives A382738.
Cf. A371761, A382734, A382736.
Cf. A382673, A382741.

Programs

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

Formula

E.g.f.: 1 / (exp(x) + exp(y) - exp(x+y))^3.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * Stirling2(n,j) * Stirling2(k,j).

A382674 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^4.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 77, 29, 1, 1, 61, 325, 325, 61, 1, 1, 125, 1181, 2357, 1181, 125, 1, 1, 253, 3973, 13621, 13621, 3973, 253, 1, 1, 509, 12797, 69269, 118061, 69269, 12797, 509, 1, 1, 1021, 40165, 326005, 862261, 862261, 326005, 40165, 1021, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2025

Keywords

Examples

			Square array begins:
  1,   1,    1,     1,      1,       1, ...
  1,   5,   13,    29,     61,     125, ...
  1,  13,   77,   325,   1181,    3973, ...
  1,  29,  325,  2357,  13621,   69269, ...
  1,  61, 1181, 13621, 118061,  862261, ...
  1, 125, 3973, 69269, 862261, 8712245, ...
  ...

Crossrefs

Columns k=0..2 give A000012, A036563(n+2), A382677.
Main diagonal gives A382678.
Cf. A099594, A136126, A382673.
Cf. A382736.

Programs

  • PARI
    a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2));

Formula

E.g.f.: exp(x+y) / (exp(x) + exp(y) - exp(x+y))^4.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+3,3) * Stirling2(n+1,j+1) * Stirling2(k+1,j+1).

A382676 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n+1,k+1)^2.

Original entry on oeis.org

1, 4, 52, 1372, 60316, 3964684, 363503932, 44280657292, 6913081723516, 1345238707327564, 319137578070718012, 90648956570718822412, 30369040605677566161916, 11848724306426305222109644, 5325560174867275152102351292, 2731649923185995549312271694732
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2025

Keywords

Crossrefs

Main diagonal of A382673.
Cf. A048163, A092552, A382678.

Programs

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

Formula

a(n) = (n!)^2 * [(x*y)^n] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

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)|.

A382675 a(n) = 4 - 15 * 2^n + 12 * 3^n.

Original entry on oeis.org

1, 10, 52, 208, 736, 2440, 7792, 24328, 74896, 228520, 693232, 2095048, 6315856, 19009000, 57149872, 171695368, 515577616, 1547715880, 4645113712, 13939273288, 41825684176, 125492781160, 376509800752, 1129592316808, 3388902779536, 10166959996840
Offset: 0

Views

Author

Seiichi Manyama, Apr 03 2025

Keywords

Crossrefs

Column k=2 of A382673.
Row n=2 of A382673.

Programs

  • PARI
    a(n) = 4-15*2^n+12*3^n;
Showing 1-5 of 5 results.

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