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A123610 Triangle, read by rows, where T(n,k) = (1/n)*Sum_{d|(n,k)} phi(d) * binomial(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).

MATHEMATICA

T[_, 0] = 1; T[n_, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]* Binomial[n/#, k/#]^2, 0]&]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Dec 06 2015, adapted from PARI *)

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

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Last modified December 10 15:32 EST 2016. Contains 279003 sequences.