login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A123610 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. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A variant of the triangle A047996 of circular binomial coefficients.

LINKS

Paul D. Hanna, Rows n = 0..45, flattened.

FORMULA

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)!.

EXAMPLE

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).

PROG

(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)))}

CROSSREFS

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 A229556

Adjacent sequences:  A123607 A123608 A123609 * A123611 A123612 A123613

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 03 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 28 17:44 EST 2014. Contains 250367 sequences.