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%I #34 Mar 11 2022 07:47:31
%S 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,
%T 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,
%U 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
%N Triangle T(n,k) read by rows: fountains of n coins and height k.
%C See A005169 for the definition of a "fountain of n coins". [_John W. Layman_, Mar 10 2011]
%H Seiichi Manyama, <a href="/A187080/b187080.txt">Rows n = 0..25, flattened</a>
%F T(n,1) + T(n,2) = Fibonacci(n).
%e Triangle begins:
%e 1;
%e 0,1;
%e 0,1,0;
%e 0,1,1,0;
%e 0,1,2,0,0;
%e 0,1,4,0,0,0;
%e 0,1,7,1,0,0,0;
%e 0,1,12,2,0,0,0,0;
%e 0,1,20,5,0,0,0,0,0;
%e 0,1,33,11,0,0,0,0,0,0;
%e 0,1,54,22,1,0,0,0,0,0,0;
%e 0,1,88,44,2,0,0,0,0,0,0,0;
%e 0,1,143,85,5,0,0,0,0,0,0,0,0;
%e 0,1,232,161,12,0,0,0,0,0,0,0,0,0;
%e 0,1,376,302,25,0,0,0,0,0,0,0,0,0,0;
%e 0,1,609,559,52,1,0,0,0,0,0,0,0,0,0,0;
%e 0,1,986,1026,105,2,0,0,0,0,0,0,0,0,0,0,0;
%e 0,1,1596,1870,207,5,0,0,0,0,0,0,0,0,0,0,0,0;
%e The 15 compositions corresponding to fountains of 7 coins are the following:
%e #: composition height
%e 1: [ 1 2 3 1 ] 3
%e 2: [ 1 2 2 2 ] 2
%e 3: [ 1 1 2 3 ] 3
%e 4: [ 1 2 2 1 1 ] 2
%e 5: [ 1 2 1 2 1 ] 2
%e 6: [ 1 1 2 2 1 ] 2
%e 7: [ 1 2 1 1 2 ] 2
%e 8: [ 1 1 2 1 2 ] 2
%e 9: [ 1 1 1 2 2 ] 2
%e 10: [ 1 2 1 1 1 1 ] 2
%e 11: [ 1 1 2 1 1 1 ] 2
%e 12: [ 1 1 1 2 1 1 ] 2
%e 13: [ 1 1 1 1 2 1 ] 2
%e 14: [ 1 1 1 1 1 2 ] 2
%e 15: [ 1 1 1 1 1 1 1 ] 1
%e stats: 0 1 12 2 0 0 0 0
%t 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 T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, 0, 0];
%t Table[T[n], {n, 0, 25}] // Flatten (* _Jean-François Alcover_, May 31 2019, after _Alois P. Heinz_ in A291878 *)
%Y Row sums give A005169 (fountains of n coins).
%Y Cf. A047998, A187081 (sandpiles by height).
%K nonn,tabl
%O 0,13
%A _Joerg Arndt_, Mar 08 2011
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