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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. 12
 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, 1, 1, 7, 46, 281, 1564, 7553, 26830, 1, 1, 8, 61, 441, 2993, 18679, 101961, 412015, 1, 1, 9, 78, 649, 5156, 38705, 268714, 1639529, 7433032, 1, 1, 10, 97, 911, 8257, 71801, 592489, 4538209, 30640257, 154076201, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS See A088956 for the definition of the hyperbinomial transform. 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 LINKS Alois P. Heinz, Rows n = 0..140, flattened N. J. A. Sloane, Transforms FORMULA E.g.f. of column k: exp(x) * (-LambertW(-x)/x)^k. A(n,k) = Sum_{j=0..n} k * (n-j+k)^(n-j-1) * C(n,j). EXAMPLE Square array begins:   1,     1,      1,      1,      1,       1,       1, ...   1,     2,      3,      4,      5,       6,       7, ...   1,     6,     13,     22,     33,      46,      61, ...   1,    29,     81,    163,    281,     441,     649, ...   1,   212,    689,   1564,   2993,    5156,    8257, ...   1,  2117,   7553,  18679,  38705,   71801,  123217, ...   1, 26830, 101961, 268714, 592489, 1166886, 2120545, ... MAPLE 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: A:= hymtr(1): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA 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 *) CROSSREFS Columns m=0-10 give: A000012, A088957, A089461, A089464, A218496, A218497, A218498, A218499, A218500, A218501, A218502. Rows n=0-2 give: A000012, A000027, A028872. Main diagonal gives A252766. Cf. A007318, A088956. Sequence in context: A128325 A307883 A111528 * A287024 A107702 A174480 Adjacent sequences:  A144300 A144301 A144302 * A144304 A144305 A144306 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 17 2008, revised Oct 30 2012 STATUS approved

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Last modified May 12 02:06 EDT 2021. Contains 343808 sequences. (Running on oeis4.)