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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.
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%I #11 Dec 06 2020 06:26:37

%S 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,

%T 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,

%U 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

%N 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.

%C 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).

%C 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.

%e The triangle starts for n>=1, 1<=k<=n (row sums after semicolons) as

%e 1 ; 1

%e 0 1 ; 1

%e 1 0 1 ; 2

%e 0 2 0 1 ; 3

%e 1 0 3 0 1 ; 5

%e 1 3 0 4 0 1 ; 9

%e 1 2 6 0 5 0 1 ; 15

%e 2 4 3 10 0 6 0 1 ; 26

%e 2 6 10 4 15 0 7 0 1 ; 45

%e 4 7 12 20 5 21 0 8 0 1 ; 78

%e 4 14 18 20 35 6 28 0 9 0 1 ; 135

%e 7 15 33 39 30 56 7 36 0 10 0 1 ; 234

%e 9 28 39 64 75 42 84 8 45 0 11 0 1 ; 406

%e 13 35 75 86 110 132 56 120 9 55 0 12 0 1 ; 704

%e 19 56 94 164 171 174 217 72 165 10 66 0 13 0 1 ; 1222

%e 25 80 162 212 315 315 259 338 90 220 11 78 0 14 0 1 ; 2120

%e 38 114 228 384 430 552 546 368 504 110 286 12 91 0 15 0 1 ; 3679

%e 51 174 349 538 800 810 903 900 504 725 132 364 13 105 0 16 0 1 ; 6385

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

%K nonn,tabl,easy

%O 1,8

%A _R. J. Mathar_, Aug 12 2020