A308292 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} 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, 2, 1, 1, 5, 3, 1, 1, 16, 19, 4, 1, 1, 65, 271, 69, 5, 1, 1, 326, 7365, 5248, 251, 6, 1, 1, 1957, 326011, 1107697, 110251, 923, 7, 1, 1, 13700, 21295783, 492911196, 191448941, 2435200, 3431, 8, 1, 1, 109601, 1924223799, 396643610629, 904434761801, 35899051101, 55621567, 12869, 9, 1

Offset

0,5

Comments

For r > 1, row r is asymptotic to sqrt(2*Pi) * (r*n)^(r*n + 1/2) / ((r!)^n * exp(r*n-1)). - Vaclav Kotesovec, May 24 2020

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 = 69.
Square array begins:
   1, 1,    1,        1,             1,                   1, ...
   1, 2,    5,       16,            65,                 326, ...
   1, 3,   19,      271,          7365,              326011, ...
   1, 4,   69,     5248,       1107697,           492911196, ...
   1, 5,  251,   110251,     191448941,        904434761801, ...
   1, 6,  923,  2435200,   35899051101,    1856296498826906, ...
   1, 7, 3431, 55621567, 7101534312685, 4098746255797339511, ...

Crossrefs

Columns k=0..4 give A000012, A000027(n+1), A030662(n+1), A144660, A144661.

Rows n=0..4 give A000012, A000522, A003011, A308294, A308295.

Main diagonal gives A274762.

Cf. A144510.

Sequence in context: A078920 A186020 A241579 * A117396 A125860 A294585

Adjacent sequences: A308289 A308290 A308291 * A308293 A308294 A308295

Keyword

nonn,tabl

Author

Seiichi Manyama, May 19 2019

Status

approved