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%I #43 May 24 2020 10:22:12
%S 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,
%T 6,1,1,1957,326011,1107697,110251,923,7,1,1,13700,21295783,492911196,
%U 191448941,2435200,3431,8,1,1,109601,1924223799,396643610629,904434761801,35899051101,55621567,12869,9,1
%N 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.
%C 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
%H Seiichi Manyama, <a href="/A308292/b308292.txt">Antidiagonals n = 0..50, flattened</a>
%F 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.
%e 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.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 5, 16, 65, 326, ...
%e 1, 3, 19, 271, 7365, 326011, ...
%e 1, 4, 69, 5248, 1107697, 492911196, ...
%e 1, 5, 251, 110251, 191448941, 904434761801, ...
%e 1, 6, 923, 2435200, 35899051101, 1856296498826906, ...
%e 1, 7, 3431, 55621567, 7101534312685, 4098746255797339511, ...
%Y Columns k=0..4 give A000012, A000027(n+1), A030662(n+1), A144660, A144661.
%Y Rows n=0..4 give A000012, A000522, A003011, A308294, A308295.
%Y Main diagonal gives A274762.
%Y Cf. A144510.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, May 19 2019
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