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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.

%I A337048 #11 Dec 06 2020 06:26:37
%S A337048 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 A337048 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 A337048 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 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.
%C A337048 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 A337048 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 A337048 The triangle starts for n>=1, 1<=k<=n (row sums after semicolons) as
%e A337048    1 ; 1
%e A337048    0   1 ; 1
%e A337048    1   0   1 ; 2
%e A337048    0   2   0   1 ; 3
%e A337048    1   0   3   0   1 ; 5
%e A337048    1   3   0   4   0   1 ; 9
%e A337048    1   2   6   0   5   0   1 ; 15
%e A337048    2   4   3  10   0   6   0   1 ; 26
%e A337048    2   6  10   4  15   0   7   0   1 ; 45
%e A337048    4   7  12  20   5  21   0   8   0   1 ; 78
%e A337048    4  14  18  20  35   6  28   0   9   0   1 ; 135
%e A337048    7  15  33  39  30  56   7  36   0  10   0   1 ; 234
%e A337048    9  28  39  64  75  42  84   8  45   0  11   0   1 ; 406
%e A337048   13  35  75  86 110 132  56 120   9  55   0  12   0   1 ; 704
%e A337048   19  56  94 164 171 174 217  72 165  10  66   0  13   0   1 ; 1222
%e A337048   25  80 162 212 315 315 259 338  90 220  11  78   0  14   0   1 ; 2120
%e A337048   38 114 228 384 430 552 546 368 504 110 286  12  91   0  15   0   1 ; 3679
%e A337048   51 174 349 538 800 810 903 900 504 725 132 364  13 105   0  16   0   1 ; 6385
%Y A337048 Cf. A005169 (row sums), A291148 (k=1, reversed sign).
%K A337048 nonn,tabl,easy
%O A337048 1,8
%A A337048 _R. J. Mathar_, Aug 12 2020