A332700 A(n, k) = Sum_{j=0..n} j!*Stirling2(n, j)*(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 13, 4, 1, 1, 1, 120, 75, 22, 5, 1, 1, 1, 720, 541, 160, 33, 6, 1, 1, 1, 5040, 4683, 1456, 285, 46, 7, 1, 1, 1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1, 1, 362880, 545835, 202672, 40005, 5656, 679, 78, 9, 1, 1

Offset

0,8

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150 of the square, flattened).

Formula

A(n, k) = Sum_{j=0..n} E(n, j)*k^j, where E(n, k) = A173018(n, k).

A(n, 1) = n!*[x^n] 1/(1-x).

A(n, k) = n!*[x^n] (k-1)/(k - exp((k-1)*x)) for k != 1.

A(n, k) = PolyLog(-n, 1/k)*(k-1)^(n+1)/k for k >= 2.

Example

Array begins:
[0] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[1] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[2] 1, 2,       3,       4,        5,         6,         7, ...    A000027
[3] 1, 6,       13,      22,       33,        46,        61, ...   A028872
[4] 1, 24,      75,      160,      285,       456,       679, ...
[5] 1, 120,     541,     1456,     3081,      5656,      9445, ...
[6] 1, 720,     4683,    15904,    40005,     84336,     158095, ...
[7] 1, 5040,    47293,   202672,   606033,    1467376,   3088765, ...
[8] 1, 40320,   545835,  2951680,  10491885,  29175936,  68958295, ...
[9] 1, 362880,  7087261, 48361216, 204343641, 652606336, 1731875605, ...
       A000142, A000670, A122704,  A255927,   A326324, ...
Seen as a triangle:
[0] [1]
[1] [1, 1]
[2] [1, 1,     1]
[3] [1, 2,     1,     1]
[4] [1, 6,     3,     1,     1]
[5] [1, 24,    13,    4,     1,    1]
[6] [1, 120,   75,    22,    5,    1,   1]
[7] [1, 720,   541,   160,   33,   6,   1,  1]
[8] [1, 5040,  4683,  1456,  285,  46,  7,  1, 1]
[9] [1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1]

Maple

# Prints array by row.
A := (n, k) -> add(combinat:-eulerian1(n, j)*k^j, j=0..n):
seq(print(seq(A(n, k), k=0..10)), n=0..8);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(n!*coeff(ser(k), x, n), k=0..9) od;
# Or:
A := (n, k) -> if k = 0 or n = 0 then 1 elif k = 1 then n! else
polylog(-n, 1/k)*(k-1)^(n+1)/k fi:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;

Mathematica

A332700[n_, k_] := n! + Sum[j! StirlingS2[n, j] (k-1)^(n-j), {j, n-1}];
Table[A332700[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 01 2024 *)

Prog

(Sage)
def T(n, k):
    return sum(factorial(j)*stirling_number2(n, j)*(k-1)^(n-j) for j in range(n+1))
for n in range(8): print([T(n, k) for k in range(8)])

Crossrefs

The matrix transpose of A326323.

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872.

Sequence in context: A139329 A335432 A229557 * A256268 A213275 A069777

Adjacent sequences: A332697 A332698 A332699 * A332701 A332702 A332703

Keyword

nonn,tabl

Author

Peter Luschny, Feb 28 2020

Status

approved