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A144304 Square array A(n,m), n>=0, m>=0, read by antidiagonals: A(n,m) = n-th number of the m-th iteration of the hyperbinomial transform on sequence A001858.

%I A144304 #21 Sep 14 2018 16:30:16
%S A144304 1,1,1,1,2,2,1,3,7,7,1,4,14,38,38,1,5,23,93,291,291,1,6,34,178,822,
%T A144304 2932,2932,1,7,47,299,1763,9193,36961,36961,1,8,62,462,3270,21504,
%U A144304 125292,561948,561948,1,9,79,673,5523,43135,313585,2022555,10026505,10026505,1
%N A144304 Square array A(n,m), n>=0, m>=0, read by antidiagonals: A(n,m) = n-th number of the m-th iteration of the hyperbinomial transform on sequence A001858.
%H A144304 Alois P. Heinz, <a href="/A144304/b144304.txt">Antidiagonals n = 0..140, flattened</a>
%H A144304 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%e A144304 Square array begins:
%e A144304    1,   1,   1,    1,    1, ...
%e A144304    1,   2,   3,    4,    5, ...
%e A144304    2,   7,  14,   23,   34, ...
%e A144304    7,  38,  93,  178,  299, ...
%e A144304   38, 291, 822, 1763, 3270, ...
%p A144304 hymtr:= proc(p) proc(n,m) `if`(m=0, p(n), m*add(p(k) *binomial(n, k) *(n-k+m)^(n-k-1), k=0..n)) end end: f:= proc(n) option remember; add(add(binomial(m, j) *binomial(n-1, n-m-j) *n^(n-m-j) *(m+j)!/ (-2)^j/ m!, j=0..m), m=0..n) end: A:= hymtr(f): seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A144304 hymtr[p_] := Function[{n, m}, If[m == 0, p[n], m*Sum[p[k]*Binomial[n, k]*(n-k+m)^(n-k-1), {k, 0, n}]]]; f[0] = 1; f[n_] := f[n] = Sum[Sum[Binomial[m, j]*Binomial[n-1, n-m-j]*n^(n-m-j)*(m+j)!/(-2)^j/m!, {j, 0, m}], {m, 0, n}]; A[0, _] = 1; A[1, k_] := k+1; A[n_, m_] := hymtr[f][n, m]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A144304 Columns m=0-3 give: A001858, A001858(n+1), A089462, A089465.
%Y A144304 Rows n=0-2 give: A000012, A000027, A008865(m+2).
%Y A144304 Main diagonal gives A252727.
%K A144304 nonn,tabl
%O A144304 0,5
%A A144304 _Alois P. Heinz_, Sep 17 2008