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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. Petros Hadjicostas, Proofs of some formulae for g.f.'s of this sequence 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)!. From Petros Hadjicostas, Oct 24 2017: (Start) Proofs of the following formulae can be found in the links. G.f.: Sum_{n>=1, k>=0} T(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(f(x^s,y^s)), where phi(s) is Euler's totient function at s, f(x,y) = (sqrt(g(x,y))+1-(1+y)*x)/2, and g(x,y) = 1-2*(1+y)*x+(1-y)^2*x^2. (Term T(0,0) is not used in this g.f.) Row g.f.: Sum_{k>=0} T(n,k)*y^k = (1/n)*Sum_{d|n} phi(d)*R(n/d, y^d), where R(m, y) = [z^m](1+(1+y)*z+y*z^2)^m. (End) 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-François 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: A297020 A099597 A283113 * 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 March 22 07:42 EDT 2018. Contains 301047 sequences. (Running on oeis4.)