OFFSET
0,12
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1274
EXAMPLE
Array begins:
=======================================================
n\k| 1 2 3 4 5 6
---+---------------------------------------------------
0 | 1 1 1 1 1 1 ...
1 | 1 1 1 2 3 6 ...
2 | 1 1 4 21 111 604 ...
3 | 1 2 22 494 9400 157040 ...
4 | 1 3 201 18086 1141055 55967596 ...
5 | 1 6 2244 794696 161927208 23276467936 ...
6 | 1 12 29096 38695548 25334545270 10673231900808 ...
...
MATHEMATICA
nmax = 10; seq[n_, k_] := Module[{p, q, h, c}, p = 1 + InverseSeries[ x/(k*2^(k - 3)*(1 + x)^k) + O[x]^n, x]; h = p /. x -> x^2 + O[x]^n; q = x*D[p, x]/p; c = Integrate[((p - 1)/k + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x] + If[OddQ[k], 0, 2^(k/2 - 2)*x*h^(k/2)]; If[k == 1, 2/(1 - x) + O[x]^n, 3/2 + c + If[OddQ[k], h + x^2*2^(k - 3)*h^k + x*2^((k - 1)/2)*h^((k + 1)/2), If[k == 2, x*h, 0] + h/(1 - 2^(k/2 - 1)*x*h^(k/2))]/2]/2];
Clear[col]; col[k_] := col[k] = CoefficientList[seq[nmax, k], x];
T[n_, k_] := col[k][[n + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 04 2018, after Andrew Howroyd *)
PROG
(PARI)
seq(n, k)={ \\ gives gf of k'th column
my(p=1 + serreverse( x/(k*2^(k-3)*(1 + x)^k) + O(x*x^n) ));
my(h=subst(p, x, x^2+O(x*x^n)), q=x*deriv(p)/p);
my(c=intformal( ((p-1)/k + sum(d=2, n, eulerphi(d)*subst(q, x, x^d+O(x*x^n))))/x) + if(k%2, 0, 2^(k/2-2)*x*h^(k/2)));
if(k==1, 2/(1-x) + O(x*x^n), 3/2 + c + if(k%2, h + x^2*2^(k-3)*h^k + x*2^((k-1)/2)*h^((k+1)/2), if(k==2, x*h, 0) + h/(1-2^(k/2-1)*x*h^(k/2)) )/2)/2;
}
Mat(vector(6, k, Col(seq(7, k))))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, May 01 2018
STATUS
approved