A308322 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..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, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -2, 3, 0, 1, 1, 9, 37, 9, 1, 1, 1, -44, 997, -692, 31, 0, 1, 1, 265, 44121, 148041, 14371, 111, 1, 1, 1, -1854, 2882071, -66211704, 25413205, -315002, 407, 0, 1, 1, 14833, 260415373, 53414037505, 120965241901, 4744544613, 7156969, 1513, 1, 1

Offset

0,12

Links

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

Formula

A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * (-x)^i/i! = (Sum_{i=0..n} x^i/i!)^k.

Example

For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + (-2)*(-x) + 4*(-x)^2/2 + (-8)*(-x)^3/6 + 14*(-x)^4/24 + (-20)*(-x)^5/120 + 20*(-x)^6/720. So A(3,2) = 1 - 2 + 4 - 8 + 14 - 20 + 20 = 9.
Square array begins:
   1, 1,   1,       1,            1,                  1, ...
   1, 0,   1,      -2,            9,                -44, ...
   1, 1,   3,      37,          997,              44121, ...
   1, 0,   9,    -692,       148041,          -66211704, ...
   1, 1,  31,   14371,     25413205,       120965241901, ...
   1, 0, 111, -315002,   4744544613,   -247578134832564, ...
   1, 1, 407, 7156969, 935728207597, 545591130328772081, ...

Crossrefs

Columns k=0..5 give A000012, A059841, A120305, A307318, A307324, A308325.

Rows n=0..1 give A000012, A182386.

Main diagonal gives A308323.

Cf. A308292.

Sequence in context: A377666 A336201 A271369 * A308898 A106728 A292603

Adjacent sequences: A308319 A308320 A308321 * A308323 A308324 A308325

Keyword

sign,tabl

Author

Seiichi Manyama, May 20 2019

Status

approved