0,5

A variant of the triangle A047996 of circular binomial coefficients. 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)!.

T(2n+1,n) = (2n+1)*A000108(n)^2 = (2n+1)*( (2n)!/(n!(n+1)!) )^2 = A000891(n) for n>=0. Surprisingly, row sums are 2*A047996(2*n,n) = 2*A003239(n) for n>0.

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).

(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)))}

Columns: A005997, A123613, A123614, A123615, A123616; A123611 (row sums), A123612 (antidiagonal sums), central terms: A123617, A123618, A123619; A047996 (variant).

Sequence in context: A114202 A125806 A099597 this_sequence A059922 A137896 A054450

Adjacent sequences: A123607 A123608 A123609 this_sequence A123611 A123612 A123613

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 03 2006