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 A187080 Triangle T(n,k) read by rows: fountains of n coins and height k. 2
 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 1, 12, 2, 0, 0, 0, 0, 0, 1, 20, 5, 0, 0, 0, 0, 0, 0, 1, 33, 11, 0, 0, 0, 0, 0, 0, 0, 1, 54, 22, 1, 0, 0, 0, 0, 0, 0, 0, 1, 88, 44, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 143, 85, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 232, 161, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 376, 302, 25, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS See A005169 for the definition of a "fountain of n coins".  [John W. Layman, Mar 10 2011] LINKS Seiichi Manyama, Rows n = 0..25, flattened FORMULA T(n,1)+T(n,2) = Fibonacci(n). EXAMPLE Triangle begins: 1; 0,1; 0,1,0; 0,1,1,0; 0,1,2,0,0; 0,1,4,0,0,0; 0,1,7,1,0,0,0; 0,1,12,2,0,0,0,0; 0,1,20,5,0,0,0,0,0; 0,1,33,11,0,0,0,0,0,0; 0,1,54,22,1,0,0,0,0,0,0; 0,1,88,44,2,0,0,0,0,0,0,0; 0,1,143,85,5,0,0,0,0,0,0,0,0; 0,1,232,161,12,0,0,0,0,0,0,0,0,0; 0,1,376,302,25,0,0,0,0,0,0,0,0,0,0; 0,1,609,559,52,1,0,0,0,0,0,0,0,0,0,0; 0,1,986,1026,105,2,0,0,0,0,0,0,0,0,0,0,0; 0,1,1596,1870,207,5,0,0,0,0,0,0,0,0,0,0,0,0; The 15 compositions corresponding to fountains of 7 coins are the following:    #:    composition      height    1:    [ 1 2 3 1 ]        3    2:    [ 1 2 2 2 ]        2    3:    [ 1 1 2 3 ]        3    4:    [ 1 2 2 1 1 ]      2    5:    [ 1 2 1 2 1 ]      2    6:    [ 1 1 2 2 1 ]      2    7:    [ 1 2 1 1 2 ]      2    8:    [ 1 1 2 1 2 ]      2    9:    [ 1 1 1 2 2 ]      2   10:    [ 1 2 1 1 1 1 ]    2   11:    [ 1 1 2 1 1 1 ]    2   12:    [ 1 1 1 2 1 1 ]    2   13:    [ 1 1 1 1 2 1 ]    2   14:    [ 1 1 1 1 1 2 ]    2   15:    [ 1 1 1 1 1 1 1 ]  1   stats:  0 1 12 2 0 0 0 0 MATHEMATICA b[n_, i_, h_] := b[n, i, h] = If[n == 0, x^h, Sum[b[n - j, j, Max[h, j]], {j, 1, Min[i + 1, n]}]]; T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, 0, 0]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, May 31 2019, after Alois P. Heinz in A291878 *) CROSSREFS Row sums give A005169 (fountains of n coins). Cf. A047998, A187081 (sand piles by height). Sequence in context: A227345 A123262 A191906 * A301342 A226369 A320751 Adjacent sequences:  A187077 A187078 A187079 * A187081 A187082 A187083 KEYWORD nonn,tabl AUTHOR Joerg Arndt, Mar 08 2011 STATUS approved

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Last modified October 23 20:01 EDT 2019. Contains 328373 sequences. (Running on oeis4.)