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 A246573 G.f.: sqrt( Sum_{n>=0} x^n / (1-x)^(4*n+3) * [Sum_{k=0..2*n+1} C(2*n+1,k)^2 * x^k]^2 ). 4

%I #5 Aug 30 2014 21:13:21

%S 1,3,15,125,1033,9385,88531,858739,8517503,85867417,877145957,

%T 9056393207,94337137375,990036525507,10456495695277,111048017798677,

%U 1185005002916425,12698840465721691,136594962042690591,1474203992211840997,15958236903892529399,173216891100594266403

%N G.f.: sqrt( Sum_{n>=0} x^n / (1-x)^(4*n+3) * [Sum_{k=0..2*n+1} C(2*n+1,k)^2 * x^k]^2 ).

%C Self-convolution equals A246571.

%e G.f.: A(x) = 1 + 3*x + 15*x^2 + 125*x^3 + 1033*x^4 + 9385*x^5 + 88531*x^6 +...

%e such that

%e A(x)^2 = 1/(1-x)^3 * (1 + x)^2 + x/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3)^2

%e + x^2/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2

%e + x^3/(1-x)^15 * (1 + 7^2*x + 21^2*x^2 + 35^2*x^3 + 35^2*x^4 + 21^2*x^5 + 7^2*x^6 + x^7)^2 +...

%e Explicitly,

%e A(x)^2 = 1 + 6*x + 39*x^2 + 340*x^3 + 3041*x^4 + 28718*x^5 + 279987*x^6 +...+ A246571(n)*x^n +...

%o (PARI) /* By definition: */

%o {a(n)=local(A=1); A = sqrt( sum(m=0, n, x^m/(1-x)^(4*m+3) * sum(k=0, 2*m+1, binomial(2*m+1, k)^2 * x^k)^2 +x*O(x^n)) ); polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A246563, A246570, A246571, A246572.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 30 2014

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Last modified August 9 13:53 EDT 2024. Contains 375042 sequences. (Running on oeis4.)