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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.
0
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
1,8
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.
EXAMPLE
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).
Sequence in context: A228783 A230425 A230426 * A318051 A301569 A301568
KEYWORD
nonn,tabl,easy
AUTHOR
R. J. Mathar, Aug 12 2020
STATUS
approved