A193050 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A003059(n+1), where A003059 is defined by "n appears 2n-1 times.".
1, 1, 1, 2, 4, 8, 17, 38, 87, 204, 489, 1191, 2938, 7328, 18448, 46809, 119583, 307324, 793965, 2060770, 5371156, 14051901, 36887289, 97131351, 256488187, 679046184, 1802047427, 4792800096, 12773166908, 34106055493, 91228795961, 244427136822, 655900969465
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 38*x^7 +... The g.f. satisfies: 1 = A(-x) + x*A(-x)^2 + x^2*A(-x)^2 + x^3*A(-x)^2 + x^4*A(-x)^3 + x^5*A(-x)^3 + x^6*A(-x)^3 + x^7*A(-x)^3 + x^8*A(-x)^3 + x^9*A(-x)^4 +...+ x^n*A(-x)^A003059(n+1) +... where A003059 begins: [1, 2,2,2, 3,3,3,3,3, 4,4,4,4,4,4,4, 5,...]. The g.f. also satisfies: 1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^2 + x^4*(1-x^5)*A(-x)^3 + x^9*(1-x^7)*A(-x)^4 + x^16*(1-x^9)*A(-x)^5 +...
Programs
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PARI
{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}
Formula
G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1 - x^(2*n-1)) * A(-x)^n.
Comments