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
Examples
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, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..50, flattened
Crossrefs
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.