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A226031 Number A(n,k) of unimodal functions f:[n]->[k*n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A226031 #25 Dec 20 2020 08:13:22
%S A226031 1,1,0,1,1,0,1,2,4,0,1,3,16,22,0,1,4,36,161,130,0,1,5,64,525,1716,791,
%T A226031 0,1,6,100,1222,8086,18832,4900,0,1,7,144,2360,24616,128248,210574,
%U A226031 30738,0,1,8,196,4047,58730,510664,2072862,2385644,194634,0
%N A226031 Number A(n,k) of unimodal functions f:[n]->[k*n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A226031 Alois P. Heinz, <a href="/A226031/b226031.txt">Antidiagonals n = 0..140, flattened</a>
%F A226031 A(n,k) = Sum_{j=0..k*n-1} C(n+2*j-1,2*j), A(0,k) = 1.
%F A226031 A(n,k) = A071921(n,k*n).
%e A226031 Square array A(n,k) begins:
%e A226031   1,   1,     1,      1,      1,       1, ...
%e A226031   0,   1,     2,      3,      4,       5, ...
%e A226031   0,   4,    16,     36,     64,     100, ...
%e A226031   0,  22,   161,    525,   1222,    2360, ...
%e A226031   0, 130,  1716,   8086,  24616,   58730, ...
%e A226031   0, 791, 18832, 128248, 510664, 1505205, ...
%p A226031 A:= (n, k)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..k*n-1)):
%p A226031 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A226031 A[n_, k_] := If[n==0, 1, Sum[Binomial[n + 2j - 1, 2j], {j, 0, k n - 1}]];
%t A226031 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ *)
%Y A226031 Columns k=0-2 give: A000007, A088536, A226012.
%Y A226031 Rows n=0-2 give: A000012, A001477, A016742.
%Y A226031 Main diagonal gives: A227402.
%Y A226031 Cf. A071920.
%K A226031 nonn,tabl
%O A226031 0,8
%A A226031 _Alois P. Heinz_, May 23 2013