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 A212326 G.f. satisfies: A(x) = theta_3( x*A(x) )^2, where theta_3(x) is Jacobi's theta_3 function. 0
 1, 4, 20, 112, 676, 4312, 28704, 197600, 1397060, 10090676, 74152456, 552666448, 4167528000, 31736182776, 243698432960, 1884809367456, 14668777816708, 114789815231560, 902661488046900, 7129068237647408, 56524456978032904, 449752267499647104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA G.f. A(x) satisfies: (1) A(x) = (1 + 2*Sum_{n>=1} (x*A(x))^(n^2) )^2. (2) A(x) =  1 + 4*Sum_{n>=1} (x*A(x))^n / (1 + (x*A(x))^(2*n)). (3) A(x) = Product_{n>=1} (1 - (-x)^n*A(x)^n)^2 / (1 + (-x)^n*A(x)^n)^2. (4) A( x/theta_3(x)^2 ) = theta_3(x)^2. (5) A(x) = (1/x)*Series_Reversion(x/theta_3(x)^2), where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1). EXAMPLE G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4312*x^5 + 28704*x^6 +... Given g.f. A(x), let q = x*A(x), then by a q-series identity: A(x) = 1 + 4*q/(1+q^2) + 4*q^2/(1+q^4) + 4*q^3/(1+q^6) + 4*q^4/(1+q^8) +... A(x) = (1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 +...)^2. ... Illustrate a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1) by the following table of coefficients in powers theta_3(x)^(2*n+2) for n>=0: n=0: [(1), 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0,...]; n=1: [1, (8), 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112,...]; n=2: [1, 12, (60), 160, 252, 312, 544, 960, 1020, 876, 1560, 2400,...]; n=3: [1, 16, 112, (448), 1136, 2016, 3136, 5504, 9328, 12112,...]; n=4: [1, 20, 180, 960, (3380), 8424, 16320, 28800, 52020, 88660,...]; n=5: [1, 24, 264, 1760, 7944, (25872), 64416, 133056, 253704,...]; n=6: [1, 28, 364, 2912, 16044, 64792, (200928), 503360, ...]; n=7: [1, 32, 480, 4480, 29152, 140736, 525952, (1580800), ...]; ... where the coefficients in parenthesis form the initial terms of this sequence: A = [1/1, 8/2, 60/3, 448/4, 3380/5, 25872/6, 200928/7, 1580800/8, ...]. PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+4*sum(m=1, n, (x*A)^m/(1+(x*A+x*O(x^n))^(2*m)))); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+2*sum(m=1, sqrtint(n+1), (x*A+x*O(x^n))^(m^2)))^2); polcoeff(A, n)} (PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1-(-x)^m*A^m)/(1+(-x)^m*A^m +x*O(x^n)))^2); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A166952. Sequence in context: A080609 A003645 A081085 * A192624 A209200 A294119 Adjacent sequences:  A212323 A212324 A212325 * A212327 A212328 A212329 KEYWORD nonn AUTHOR Paul D. Hanna, May 14 2012 STATUS approved

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Last modified December 10 23:13 EST 2019. Contains 329909 sequences. (Running on oeis4.)