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%I #11 Oct 09 2019 16:38:34
%S 1,9,53,504,3479,34362,248799,2483091,18383088,185472450,1378756330,
%T 14142832704,104308903182,1093968797580,7930694023345,85563977678775,
%U 604256899812240,6755825768907204,46021002487605408,538061313239561853,3494583984824812425,43217684276354830458,263780496112409697816,3501503522404393600863,19716335477199319610336
%N G.f.: [ Sum_{n>=0} (2*n + 1) * x^n * (9 - x^n)^n ]^(1/3).
%C G.f. is congruent modulo 3 to Product_{n>=1} 1 - x^(2*n).
%C The cube of the g.f., A(x)^3, is congruent modulo 9 to Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)), and equals the g.f. of A326605.
%C First negative term is a(38) = -19995469991861952392493964610982114.
%H Paul D. Hanna, <a href="/A326606/b326606.txt">Table of n, a(n) for n = 0..400</a>
%F G.f.: [ Sum_{n>=0} (2*n+1) * x^n * (9 - x^n)^n ]^(1/3).
%F G.f.: [ Sum_{n>=0} (-1)^n * (2*n+1 + 9*x^(n+1)) * x^(n*(n+1)) / (1 - 9*x^(n+1))^(n+2) ]^(1/3).
%e G.f. A(x) = 1 + 9*x + 53*x^2 + 504*x^3 + 3479*x^4 + 34362*x^5 + 248799*x^6 + 2483091*x^7 + 18383088*x^8 + 185472450*x^9 + 1378756330*x^10 + ...
%e such that
%e A(x)^3 = 1 + 3*x*(9-x) + 5*x^2*(9-x^2)^2 + 7*x^3*(9-x^3)^3 + 9*x^4*(9-x^4)^4 + 11*x^5*(9-x^5)^5 + 13*x^6*(9-x^6)^6 + 15*x^7*(9-x^7)^7 + ...
%e Also,
%e A(x)^3 = (1 + 9*x)/(1 - 9*x)^2 - (3 + 9*x^2)*x^2/(1 - 9*x^2)^3 + (5 + 9*x^3)*x^6/(1 - 9*x^3)^4 - (7 + 9*x^4)*x^12/(1 - 9*x^4)^5 + (9 + 9*x^5)*x^20/(1 - 9*x^5)^6 - (11 + 9*x^6)*x^30/(1 - 9*x^6)^7 + (13 + 9*x^7)*x^42/(1 - 9*x^7)^8 + ...
%e RELATED SERIES.
%e G.f. A(x) is congruent modulo 3 to Product_{n>=1} 1 - x^(2*n),
%e A(x) (mod 3) = 1 - x^2 - x^4 + x^10 + x^14 - x^24 - x^30 + x^44 + x^52 - x^70 - x^80 + x^102 + x^114 - x^140 - x^154 + x^184 + x^200 + ...
%e The cube of the g.f. A(x) begins
%e A(x)^3 = 1 + 27*x + 402*x^2 + 5103*x^3 + 58959*x^4 + 649539*x^5 + 6907037*x^6 + 71744535*x^7 + 731768013*x^8 + 7360989480*x^9 + ... + A326605(n)*x^n + ...
%e which is congruent modulo 9 to
%e A(x)^3 (mod 9) = 1 - 3*x^2 + 5*x^6 - 7*x^12 + 9*x^20 - 11*x^30 + 13*x^42 - 15*x^56 + 17*x^72 - 19*x^90 + 21*x^110 + ... + (-1)^n*(2*n+1)*x^(n*(n+1)) + ...
%o (PARI) /* By definition */
%o {a(n) = my(A = sum(m=0,n, (2*m + 1) * x^m * (9 - x^m + x*O(x^n))^m)^(1/3) ); polcoeff(A,n)}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) /* Accelerated series */
%o {a(n) = my(A = sum(m=0,sqrtint(n+1), (-1)^m * (2*m + 1 + 9*x^(m+1))* x^(m*(m+1)) / (1 - 9*x^(m+1) + x*O(x^n))^(m+2) )^(1/3) ); polcoeff(A,n)}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A326605.
%K sign
%O 0,2
%A _Paul D. Hanna_, Oct 08 2019
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