OFFSET
0,2
COMMENTS
CONJECTURES. (Start)
_ The self-convolution square-root is A195443 and consists entirely of integers.
_ a(4*(2^n-1)) is odd; odd numbers occur only at positions {2^(n+2) - 4, n>=0}.
_ In the self-convolution square-root of this sequence (A195443), odd numbers occur only at positions {2^(n+1) - 2, n>=0}. (End)
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..199
EXAMPLE
G.f.: 1 = 1/(1+x)^4 + 4*x/((1+x)^4*(1+2*x)^4) + 38*x^2/((1+x)^4*(1+2*x)^4*(1+3*x)^4) + 604*x^3/((1+x)^4*(1+2*x)^4*(1+3*x)^4*(1+4*x)^4) +...
This sequence has odd terms at [0,4,12,28,60,124,...,2^(n+2)-4,...].
O.g.f.: A(x) = 1 + 4*x + 38*x^2 + 604*x^3 + 13797*x^4 + 416168*x^5 +...
where the square-root is an integer series (cf. A195443):
A(x)^(1/2) = 1 + 2*x + 17*x^2 + 268*x^3 + 6218*x^4 + 191092*x^5 + 7331943*x^6 + 338203880*x^7 + 18267488524*x^8 + 1132962942756*x^9 +...
which has odd terms at [0,2,6,14,30,62,126,...,2^(n+1)-2,...].
MATHEMATICA
a[n_] := a[n] = If[n == 0, 1, SeriesCoefficient[1-Sum[a[k] x^k/Product[1 + j x + x O[x]^n, {j, 1, k+1}]^4, {k, 0, n-1}], {x, 0, n}]];
a /@ Range[0, 16] (* Jean-François Alcover, Nov 03 2019, from PARI *)
PROG
(PARI) {a(n)=if(n==0, 1, polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, 1+j*x+x*O(x^n))^4), n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 18 2011
STATUS
approved