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%I #17 Nov 10 2019 20:42:53
%S 1,4,38,604,13797,416168,15711490,715565508,38278167466,2356801113088,
%T 164391501446500,12826066272723832,1107734575497955773,
%U 104985517080964044744,10838759916371136482170,1211294776167588947134004,145737426974188872882131694
%N G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1+k*x)^4.
%C CONJECTURES. (Start)
%C _ The self-convolution square-root is A195443 and consists entirely of integers.
%C _ a(4*(2^n-1)) is odd; odd numbers occur only at positions {2^(n+2) - 4, n>=0}.
%C _ In the self-convolution square-root of this sequence (A195443), odd numbers occur only at positions {2^(n+1) - 2, n>=0}. (End)
%H Paul D. Hanna, <a href="/A195442/b195442.txt">Table of n, a(n) for n = 0..199</a>
%e 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) +...
%e This sequence has odd terms at [0,4,12,28,60,124,...,2^(n+2)-4,...].
%e O.g.f.: A(x) = 1 + 4*x + 38*x^2 + 604*x^3 + 13797*x^4 + 416168*x^5 +...
%e where the square-root is an integer series (cf. A195443):
%e 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 +...
%e which has odd terms at [0,2,6,14,30,62,126,...,2^(n+1)-2,...].
%t 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}]];
%t a /@ Range[0, 16] (* _Jean-François Alcover_, Nov 03 2019, from PARI *)
%o (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))}
%Y Cf. A118804, A193333, A195443.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 18 2011