A308356 A(n,k) = (1/k!) * Sum_{i_1=1..n} Sum_{i_2=1..n} ... Sum_{i_k=1..n} (-1)^(i_1 + i_2 + ... + i_k) * multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

1, 0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, -1, 1, 0, 1, 5, 5, 0, 1, 0, -1, 36, -120, 15, -1, 1, 0, 1, 329, 6286, 2380, 56, 0, 1, 0, -1, 3655, -557991, 1056496, -52556, 203, -1, 1, 0, 1, 47844, 74741031, 1006985994, 197741887, 1192625, 757, 0, 1

Offset

0,18

Links

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Formula

A(n,k) = Sum_{i=k..k*n} b(i) where Sum_{i=k..k*n} b(i) * (-x)^i/i! = (1/k!) * (Sum_{i=1..n} x^i/i!)^k.

Example

For (n,k) = (3,2), (1/2) * (Sum_{i=1..3} x^i/i!)^2 = (1/2) * (x + x^2/2 + x^3/6)^2 = (-x)^2/2 + (-3)*(-x)^3/6 + 7*(-x)^4/24 + (-10)*(-x)^5/120 + 10*(-x)^6/720. So A(3,2) = 1 - 3 + 7 - 10 + 10 = 5.
Square array begins:
   1,  0,   0,       0,           0,                0, ...
   1, -1,   1,      -1,           1,               -1, ...
   1,  0,   1,       5,          36,              329, ...
   1, -1,   5,    -120,        6286,          -557991, ...
   1,  0,  15,    2380,     1056496,       1006985994, ...
   1, -1,  56,  -52556,   197741887,   -2063348839223, ...
   1,  0, 203, 1192625, 38987482590, 4546553764660831, ...

Crossrefs

Columns k=0..4 give A000012, (-1)*A000035, A307349, (-1)*A307350, A307351.

Rows n=0..5 give A000007, A033999, A278990, A308363, A308389, A308390.

Main diagonal gives A308327.

Cf. A144510.

Sequence in context: A030047 A176184 A176144 * A065936 A299625 A021649

Adjacent sequences: A308353 A308354 A308355 * A308357 A308358 A308359

Keyword

sign,tabl

Author

Seiichi Manyama, May 21 2019

Status

approved