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A144303 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 the sequence of 1's.

%I A144303 #32 Oct 04 2018 20:19:31
%S A144303 1,1,1,1,2,1,1,3,6,1,1,4,13,29,1,1,5,22,81,212,1,1,6,33,163,689,2117,
%T A144303 1,1,7,46,281,1564,7553,26830,1,1,8,61,441,2993,18679,101961,412015,1,
%U A144303 1,9,78,649,5156,38705,268714,1639529,7433032,1,1,10,97,911,8257,71801,592489,4538209,30640257,154076201,1
%N A144303 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 the sequence of 1's.
%C A144303 See A088956 for the definition of the hyperbinomial transform.
%C A144303 A(n,m), n>=0, m>=0, is the number of subtrees of the complete graph K_{n+m} on n+m vertices containing a given, fixed subtree on m vertices. - _Alex Chin_, Jul 25 2013
%H A144303 Alois P. Heinz, <a href="/A144303/b144303.txt">Rows n = 0..140, flattened</a>
%H A144303 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A144303 E.g.f. of column k: exp(x) * (-LambertW(-x)/x)^k.
%F A144303 A(n,k) = Sum_{j=0..n} k * (n-j+k)^(n-j-1) * C(n,j).
%e A144303 Square array begins:
%e A144303   1,     1,      1,      1,      1,       1,       1, ...
%e A144303   1,     2,      3,      4,      5,       6,       7, ...
%e A144303   1,     6,     13,     22,     33,      46,      61, ...
%e A144303   1,    29,     81,    163,    281,     441,     649, ...
%e A144303   1,   212,    689,   1564,   2993,    5156,    8257, ...
%e A144303   1,  2117,   7553,  18679,  38705,   71801,  123217, ...
%e A144303   1, 26830, 101961, 268714, 592489, 1166886, 2120545, ...
%p A144303 hymtr:= proc(p) proc(n,m) `if`(m=0, p(n), m*add(
%p A144303            p(k)*binomial(n, k) *(n-k+m)^(n-k-1), k=0..n))
%p A144303         end end:
%p A144303 A:= hymtr(1):
%p A144303 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A144303 a[_, 0] = 1; a[n_, k_] := Sum[k*(n - j + k)^(n - j - 1)*Binomial[n, j], {j, 0, n}]; Table[a[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jun 24 2013 *)
%Y A144303 Columns m=0-10 give: A000012, A088957, A089461, A089464, A218496, A218497, A218498, A218499, A218500, A218501, A218502.
%Y A144303 Rows n=0-2 give: A000012, A000027, A028872.
%Y A144303 Main diagonal gives A252766.
%Y A144303 Cf. A007318, A088956.
%K A144303 nonn,tabl
%O A144303 0,5
%A A144303 _Alois P. Heinz_, Sep 17 2008, revised Oct 30 2012