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%I #32 Oct 04 2018 20:19:31
%S 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 1,1,7,46,281,1564,7553,26830,1,1,8,61,441,2993,18679,101961,412015,1,
%U 1,9,78,649,5156,38705,268714,1639529,7433032,1,1,10,97,911,8257,71801,592489,4538209,30640257,154076201,1
%N 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 See A088956 for the definition of the hyperbinomial transform.
%C 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 Alois P. Heinz, <a href="/A144303/b144303.txt">Rows n = 0..140, flattened</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F E.g.f. of column k: exp(x) * (-LambertW(-x)/x)^k.
%F A(n,k) = Sum_{j=0..n} k * (n-j+k)^(n-j-1) * C(n,j).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, 7, ...
%e 1, 6, 13, 22, 33, 46, 61, ...
%e 1, 29, 81, 163, 281, 441, 649, ...
%e 1, 212, 689, 1564, 2993, 5156, 8257, ...
%e 1, 2117, 7553, 18679, 38705, 71801, 123217, ...
%e 1, 26830, 101961, 268714, 592489, 1166886, 2120545, ...
%p hymtr:= proc(p) proc(n,m) `if`(m=0, p(n), m*add(
%p p(k)*binomial(n, k) *(n-k+m)^(n-k-1), k=0..n))
%p end end:
%p A:= hymtr(1):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t 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 Columns m=0-10 give: A000012, A088957, A089461, A089464, A218496, A218497, A218498, A218499, A218500, A218501, A218502.
%Y Rows n=0-2 give: A000012, A000027, A028872.
%Y Main diagonal gives A252766.
%Y Cf. A007318, A088956.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Sep 17 2008, revised Oct 30 2012
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