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A360764
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Number T(n,k) of sets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=max(i:T(n,i)>0), read by rows.
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4
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1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 1, 4, 6, 1, 0, 1, 6, 8, 4, 0, 1, 6, 13, 9, 1, 0, 1, 8, 18, 16, 6, 0, 1, 8, 24, 29, 13, 2, 0, 1, 10, 30, 43, 29, 6, 0, 1, 10, 39, 64, 52, 19, 1, 0, 1, 12, 46, 89, 89, 42, 7, 0, 1, 12, 56, 122, 139, 85, 22, 1
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OFFSET
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0,8
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COMMENTS
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T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.
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LINKS
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EXAMPLE
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T(6,1) = 1: {[6]}.
T(6,2) = 4: {[1],[5]}, {[2],[4]}, {[1,5]}, {[2,4]}.
T(6,3) = 6: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[2],[3]}.
T(6,4) = 1: {[1],[2],[1,2]}.
Triangle T(n,k) begins:
1;
0, 1;
0, 1;
0, 1, 2;
0, 1, 2, 1;
0, 1, 4, 2;
0, 1, 4, 6, 1;
0, 1, 6, 8, 4;
0, 1, 6, 13, 9, 1;
0, 1, 8, 18, 16, 6;
0, 1, 8, 24, 29, 13, 2;
0, 1, 10, 30, 43, 29, 6;
0, 1, 10, 39, 64, 52, 19, 1;
...
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MAPLE
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h:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i), k), k=0..j))))
end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);
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MATHEMATICA
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h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]];
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i<0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]] ;
T[n_] := CoefficientList[b[n, n], x];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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