OFFSET
1,8
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
C. G. Bower, Transforms (2)
FORMULA
G.f. satisfies A(x, y)=xy+x*CIK(A(x, y))-x. Shifts up under CIK transform.
G.f. satisfies A(x, y) = x*(y - Sum_{i>0} phi(i)/i * log(1 - A(x^i, y^i))). - Michael Somos, Aug 24 2015
Sum_k T(n, k) = A032200(n). - Michael Somos, Aug 24 2015
EXAMPLE
G.f. = x^(y + x*y + x^2*(y + y^2) + x^3*(y + 2*y^2 + y^3) + x^4*(y + 4*y^2 + 3*y^3 + y^4) + ...).
n\k 1 2 3 4 5 6 7 8
--:-- -- -- -- -- -- -- --
1: 1
2: 1 0
3: 1 1 0
4: 1 2 1 0
5: 1 4 3 1 0
6: 1 6 8 4 1 0
7: 1 9 19 16 5 1 0
8: 1 12 37 46 25 6 1 0
MATHEMATICA
m = 13; A[_, _] = 0;
Do[A[x_, y_] = x (y - Sum[EulerPhi[i]/i Log[1 - A[x^i, y^i]], {i, 1, m}]) + O[x]^m + O[y]^m // Normal, {m}];
Join[{1}, Append[CoefficientList[#/y, y], 0]& /@ Rest @ CoefficientList[ A[x, y]/x, x]] // Flatten (* Jean-François Alcover, Oct 02 2019 *)
PROG
(PARI) {T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*y - x*sum(i=1, j, eulerphi(i)/i * log(1 - subst( subst( A + x * O(x^min(j, n\i)), x, x^i), y, y^i) ) )); polcoeff( polcoeff(A, n), k))}; /* Michael Somos, Aug 24 2015 */
CROSSREFS
KEYWORD
AUTHOR
Christian G. Bower, May 14 2000
STATUS
approved