A382824 -id:A382824 - cp's OEIS frontend

A382823 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)) ).

1, 1, 1, 2, 2, 2, 6, 5, 5, 6, 24, 17, 17, 17, 24, 120, 74, 69, 69, 74, 120, 720, 394, 338, 337, 338, 394, 720, 5040, 2484, 1962, 1894, 1894, 1962, 2484, 5040, 40320, 18108, 13228, 12194, 12152, 12194, 13228, 18108, 40320, 362880, 149904, 101812, 89160, 87320, 87320, 89160, 101812, 149904, 362880

Offset: 0

Seiichi Manyama, Apr 05 2025

			Square array begins:
    1,   1,    2,     6,    24,    120, ...
    1,   2,    5,    17,    74,    394, ...
    2,   5,   17,    69,   338,   1962, ...
    6,  17,   69,   337,  1894,  12194, ...
   24,  74,  338,  1894, 12152,  87320, ...
  120, 394, 1962, 12194, 87320, 696076, ...

Columns k=0..1 give A000142, A000774.
Main diagonal gives A382826.
Cf. A382824, A382825.
Cf. A099594, A379821.

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

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

Seiichi Manyama, Apr 06 2025

			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, ...

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

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)));

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)|.

A382827 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n+1,k+1)^2.

Original entry on oeis.org

1, 3, 34, 854, 37556, 2546852, 246113904, 32104625520, 5433891955968, 1157778241057152, 303197684900579712, 95717977509042032256, 35847800701044816248064, 15713483696924130220098816, 7969364997624587289470810112, 4630203661005094483980386924544

Offset: 0

Seiichi Manyama, Apr 06 2025

nonn

Main diagonal of A382824.

Cf. A092552, A382804.

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

a(n) = (n!)^2 * [(x*y)^n] 1 / ( (1-x) * (1-y) * (1 - log(1-x) * log(1-y))^2 ).

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

cp's OEIS Frontend

A382823 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)) ).

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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 ).

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PARI

Formula

A382827 a(n) = Sum_{k=0..n} k! * (k+1)! * Stirling1(n+1,k+1)^2.

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