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A187080 Triangle T(n,k) read by rows: fountains of n coins and height k.

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