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

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

Offset: 0

Author

Joerg Arndt, Mar 08 2011

Keywords

• nonn
• tabl

Comments

See A005169 for the definition of a "fountain of n coins". [John W. Layman, Mar 10 2011]

Examples

			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
		

Links

• Seiichi Manyama, Rows n = 0..25, flattened

Crossrefs

Row sums give A005169 (fountains of n coins).

Cf. A047998, A187081 (sandpiles by height).

Programs

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 *)

Formula

T(n,1) + T(n,2) = Fibonacci(n).