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 A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k). 14
 1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 1, 10, 15, 1, 0, 0, 3, 41, 52, 1, 0, 0, 1, 9, 196, 203, 1, 0, 0, 0, 4, 40, 1057, 877, 1, 0, 0, 0, 1, 10, 210, 6322, 4140, 1, 0, 0, 0, 0, 5, 30, 1176, 41393, 21147, 1, 0, 0, 0, 0, 1, 15, 175, 7273, 293608, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 49932, 2237921, 678570 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS A(n,k) is also the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box k balls are seen at the top. E.g. A(3,1)=10:   |1.| |2.| |3.| |1|2| |1|2| |1|3| |1|3| |2|3| |2|3| |1|2|3|   |23| |13| |12| |3|.| |.|3| |2|.| |.|2| |1|.| |.|1| |.|.|.|   +--+ +--+ +--+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+-+ LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened N. J. A. Sloane, Transforms FORMULA A(0,k) = 1 and A(n,k) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0. - Seiichi Manyama, Sep 28 2017 EXAMPLE Square array A(n,k) begins:    1,   1,  1,  1,  1,  1,  ...    1,   1,  0,  0,  0,  0,  ...    2,   3,  1,  0,  0,  0,  ...    5,  10,  3,  1,  0,  0,  ...   15,  41,  9,  4,  1,  0,  ...   52, 196, 40, 10,  5,  1,  ... MAPLE exptr:= proc(p) local g; g:=           proc(n) option remember; `if`(n=0, 1,              add(binomial(n-1, j-1) *p(j) *g(n-j), j=1..n))         end: end: A:= (n, k)-> exptr(i-> binomial(i, k))(n): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA Exptr[p_] := Module[{g}, g[n_] := g[n] = If[n == 0, 1, Sum[Binomial[n-1, j-1] *p[j]*g[n-j], {j, 1, n}]]; g]; A[n_, k_] := Exptr[Function[i, Binomial[i, k]]][n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *) PROG (Ruby) def ncr(n, r)   return 1 if r == 0   (n - r + 1..n).inject(:*) / (1..r).inject(:*) end def A(k, n)   ary = [1]   (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}   ary end def A145460(n)   a = []   (0..n).each{|i| a << A(i, n - i)}   ary = []   (0..n).each{|i|     (0..i).each{|j|       ary << a[i - j][j]     }   }   ary end p A145460(20) # Seiichi Manyama, Sep 28 2017 CROSSREFS Columns k=0-9 give: A000110, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459. A(2n,n) gives A029651. Cf.: A007318, A143398, A292948. Sequence in context: A099493 A088523 A222211 * A292978 A202178 A035543 Adjacent sequences:  A145457 A145458 A145459 * A145461 A145462 A145463 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Oct 10 2008 STATUS approved

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Last modified November 16 09:21 EST 2018. Contains 317268 sequences. (Running on oeis4.)