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A123610
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Triangle, read by rows, where T(n,k) = (1/n)*Sum_{d|(n,k)} phi(d) * C(n/d,k/d)^2 for n>=k>0, with T(n,0)=1 for n>=0.
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11
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1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 39, 68, 39, 6, 1, 1, 7, 63, 175, 175, 63, 7, 1, 1, 8, 100, 392, 618, 392, 100, 8, 1, 1, 9, 144, 786, 1764, 1764, 786, 144, 9, 1, 1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1, 1, 11, 275, 2475, 9900
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OFFSET
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0,5
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COMMENTS
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A variant of the triangle A047996 of circular binomial coefficients.
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LINKS
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Paul D. Hanna, Rows n = 0..45, flattened.
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FORMULA
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T(2n+1,n) = (2n+1)*A000108(n)^2 = (2n+1)*( (2n)!/(n!(n+1)!) )^2 = A000891(n) for n>=0.
Row sums are 2*A047996(2*n,n) = 2*A003239(n) for n>0.
Row sums equal the row sums of triangle A128545.
For n>=1, the g.f. of column n has the form: P_n(x)/(Product_{m=1..n)(1-x^m)^2), where P_n(x) is a polynomial with n^2 coefficients such that the sum of the coefficients is P_n(1) = (2*n-1)!.
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 10, 4, 1;
1, 5, 20, 20, 5, 1;
1, 6, 39, 68, 39, 6, 1;
1, 7, 63, 175, 175, 63, 7, 1;
1, 8, 100, 392, 618, 392, 100, 8, 1;
1, 9, 144, 786, 1764, 1764, 786, 144, 9, 1;
1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1; ...
Example of column g.f.s are:
column 1: 1/(1-x)^2;
column 2: Ser([1,1,3,1]) / ((1-x)^2*(1-x^2)^2) = g.f. of A005997;
column 3: Ser([1,2,11,26,30,26,17,6,1]) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2);
column 4: Ser([1,3,28,94,240,440,679,839,887,757,550,314,148,48,11,1]) /
((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2);
where Ser() denotes a polynomial in x with the given coefficients,
as in Ser([1,1,3,1]) = (1 + x + 3*x^2 + x^3).
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PROG
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(PARI) {T(n, k)=if(k==0, 1, (1/n)*sumdiv(n, d, if(gcd(k, d)==d, eulerphi(d)*binomial(n/d, k/d)^2, 0)))}
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CROSSREFS
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Cf. Columns: A005997, A123613, A123614, A123615, A123616.
Cf. A123611 (row sums), A123612 (antidiagonal sums), central terms: A123617.
Cf. A123618, A123619; A047996 (variant); A128545.
Sequence in context: A202756 A156354 A099597 * A209631 A059922 A159623
Adjacent sequences: A123607 A123608 A123609 * A123611 A123612 A123613
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna, Oct 03 2006
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STATUS
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approved
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