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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Table of n, a(n) for n=0..65.

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;

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

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Last modified June 25 04:52 EDT 2021. Contains 345452 sequences. (Running on oeis4.)