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A112705
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Triangle built from partial sums of Catalan numbers A000108 multiplied by powers.
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12
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1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 11, 4, 1, 1, 23, 51, 22, 5, 1, 1, 65, 275, 157, 37, 6, 1, 1, 197, 1619, 1291, 357, 56, 7, 1, 1, 626, 10067, 11497, 3941, 681, 79, 8, 1, 1, 2056, 64979, 107725, 46949, 9431, 1159, 106, 9, 1, 1, 6918, 431059, 1045948, 587621, 140681, 19303, 1821, 137, 10, 1
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OFFSET
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0,5
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COMMENTS
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The column sequences (without leading zeros) begin with A000012 (powers of 1), A112705 (partial sums Catalan), A112696-A112704, for m=0..10.
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LINKS
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FORMULA
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a(n, m) = sum(C(k)*m^k, k=0..n-m), n>m>0, with C(n):=A000108(n); a(n, n)=1; a(n, 0)=1; a(n, m)=0 if n<m.
G.f. for column m>=0 (without leading zeros): c(m*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
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EXAMPLE
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Triangle starts:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 11, 4, 1;
1, 23, 51, 22, 5, 1;
1, 65, 275, 157, 37, 6, 1;
...
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MATHEMATICA
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col[m_] := col[m] = CatalanNumber[#]*m^#& /@ Range[0, 20] // Accumulate;
T[n_, m_] := If[m == 0, 1, col[m][[n - m + 1]]];
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PROG
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(PARI) t(n, m) = if (m==0, 1, if (n==m, 1, sum(kk=0, n-m, m^kk*binomial(2*kk, kk)/(kk+1))));
tabl(nn) = {for (n=0, nn, for (m=0, n, print1(t(n, m), ", "); ); print(); ); } \\ Michel Marcus, Nov 25 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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