A337048 Triangle T(n,k) read by rows: the number of fountains of n coins composed of k inseparable fountains of coins placed side-by-side.

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 1, 1, 3, 0, 4, 0, 1, 1, 2, 6, 0, 5, 0, 1, 2, 4, 3, 10, 0, 6, 0, 1, 2, 6, 10, 4, 15, 0, 7, 0, 1, 4, 7, 12, 20, 5, 21, 0, 8, 0, 1, 4, 14, 18, 20, 35, 6, 28, 0, 9, 0, 1, 7, 15, 33, 39, 30, 56, 7, 36, 0, 10, 0, 1, 9, 28, 39, 64, 75, 42, 84, 8, 45, 0, 11, 0, 1, 13, 35, 75, 86, 110

Offset: 1

Author

R. J. Mathar, Aug 12 2020

Keywords

• nonn
• tabl
• easy

Comments

A fountain of coins is called "inseparable" here if it cannot be split into 2 fountains of coins by a vertical cut without slicing a coin. That means: inseparable fountains have "full" second rows. They are basically counted in A291148 (apart from a sign).

The ordinary generating function of column k is g(x)^k, where g(x) = x +x^3 +x5 +x^6+.. is the ordinary generating function of column k=1 and g(x) is also the INVERTi transform of A005169.

Examples

			The triangle starts for n>=1, 1<=k<=n (row sums after semicolons) as
   1 ; 1
   0   1 ; 1
   1   0   1 ; 2
   0   2   0   1 ; 3
   1   0   3   0   1 ; 5
   1   3   0   4   0   1 ; 9
   1   2   6   0   5   0   1 ; 15
   2   4   3  10   0   6   0   1 ; 26
   2   6  10   4  15   0   7   0   1 ; 45
   4   7  12  20   5  21   0   8   0   1 ; 78
   4  14  18  20  35   6  28   0   9   0   1 ; 135
   7  15  33  39  30  56   7  36   0  10   0   1 ; 234
   9  28  39  64  75  42  84   8  45   0  11   0   1 ; 406
  13  35  75  86 110 132  56 120   9  55   0  12   0   1 ; 704
  19  56  94 164 171 174 217  72 165  10  66   0  13   0   1 ; 1222
  25  80 162 212 315 315 259 338  90 220  11  78   0  14   0   1 ; 2120
  38 114 228 384 430 552 546 368 504 110 286  12  91   0  15   0   1 ; 3679
  51 174 349 538 800 810 903 900 504 725 132 364  13 105   0  16   0   1 ; 6385
		

Crossrefs

Cf. A005169 (row sums), A291148 (k=1, reversed sign).