OFFSET
0,11
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..90 from Tilman Piesk)
FORMULA
T(n,k) = (1/n)*(A091867(n,k) - A171128(n,k) + Sum_{d|gcd(n,k)} phi(d) * A171128(n/d, k/d)) for n > 0. - Andrew Howroyd, Nov 16 2017
EXAMPLE
From Andrew Howroyd, Nov 16 2017: (Start)
Triangle begins: (n >= 0, 0 <= k <= n)
1;
0, 1;
1, 0, 1;
1, 1, 0, 1;
2, 1, 2, 0, 1;
2, 3, 2, 2, 0, 1;
5, 6, 9, 4, 3, 0, 1;
6, 15, 18, 15, 5, 3, 0, 1;
15, 36, 56, 42, 29, 7, 4, 0, 1;
28, 91, 144, 142, 84, 42, 10, 4, 0, 1;
67, 232, 419, 432, 322, 152, 66, 12, 5, 0, 1;
(End)
MATHEMATICA
a91867[n_, k_] := If[k == n, 1, (Binomial[n + 1, k]/(n + 1)) Sum[Binomial[n + 1 - k, j] Binomial[n - k - j - 1, j - 1], {j, 1, (n - k)/2}]];
a2426[n_] := Sum[Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}];
a171128[n_, k_] := Binomial[n, k]*a2426[n - k];
T[0, 0] = 1;
T[n_, k_] := (1/n)*(a91867[n, k] - a171128[n, k] + Sum[EulerPhi[d]* a171128[n/d, k/d], {d, Divisors[GCD[n, k]]}]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)
PROG
(PARI)
g(x, y) = {1/sqrt((1 - (1 + y)*x)^2 - 4*x^2) - 1}
S(n)={my(A=(1-sqrt(1-4*x/(1-(y-1)*x) + O(x^(n+2))))/(2*x)-1); Vec(1+intformal((A + sum(k=2, n, eulerphi(k)*g(x^k + O(x*x^n), y^k)))/x))}
my(v=S(10)); for(n=1, #v, my(p=v[n]); for(k=0, n-1, print1(polcoeff(p, k), ", ")); print) \\ Andrew Howroyd, Nov 16 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tilman Piesk, Apr 12 2012
STATUS
approved