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1, 2, 1, 3, 2, 1, 7, 5, 2, 1, 26, 19, 7, 2, 1, 141, 104, 37, 9, 2, 1, 1034, 766, 268, 61, 11, 2, 1, 9693, 7197, 2496, 550, 91, 13, 2, 1, 111522, 82910, 28612, 6195, 982, 127, 15, 2, 1, 1528112, 1136923, 391189, 83837, 12977, 1596, 169, 17, 2, 1, 24372513, 18141867, 6230646, 1326923, 202494, 24206, 2424, 217, 19, 2, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(2+(k+j)*(k+j-1)/2-k*(k-1)/2).
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EXAMPLE
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G.f. for column 0:
1 = T(0,0)*(1-x)^2 + T(1,0)*x*(1-x)^2 + T(2,0)*x^2*(1-x)^3 + T(3,0)*x^3*(1-x)^5 + T(4,0)*x^4*(1-x)^8 + T(5,0)*x^5*(1-x)^12 +...
= 1*(1-x)^2 + 2*x*(1-x)^2 + 3*x^2*(1-x)^3 + 7*x^3*(1-x)^5 + 26*x^4*(1-x)^8 + 141*x^5*(1-x)^12 +...
G.f. for column 1:
1 = T(1,1)*(1-x)^2 + T(2,1)*x*(1-x)^3 + T(3,1)*x^2*(1-x)^5 + T(4,1)*x^3*(1-x)^8 + T(5,1)*x^4*(1-x)^12 + T(6,1)*x^5*(1-x)^17 +...
= 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 + 19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...
Triangle T begins:
1;
2,1;
3,2,1;
7,5,2,1;
26,19,7,2,1;
141,104,37,9,2,1;
1034,766,268,61,11,2,1;
9693,7197,2496,550,91,13,2,1;
111522,82910,28612,6195,982,127,15,2,1;
...
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MATHEMATICA
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max = 10;
A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];
A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];
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PROG
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(PARI) {T(n, k)=polcoeff(1-sum(j=0, n-k-1, T(j+k, k)*x^j*(1-x+x*O(x^n))^(2+(k+j)*(k+j-1)/2-k*(k-1)/2)), n-k)}
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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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