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A326559 G.f. A(x) satisfies: Sum_{n>=0} A(x^n)^n = x. 1

%I #12 Aug 14 2019 14:15:45

%S 1,0,0,-1,0,0,0,0,-1,2,0,0,0,0,0,-2,0,3,0,2,0,-4,0,0,-1,-2,-3,8,0,0,0,

%T 0,3,4,0,-7,0,-2,0,-9,0,-10,0,6,12,8,0,8,-1,-4,3,-2,0,16,-10,-6,-3,

%U -14,0,-21,0,-14,-24,50,5,-3,0,4,6,-1,0,28,0,48,12,-68,0,-17,0,-20,3,14,0,36,35,10,-3,-38,0,-34,-21,-52,-54,-16,30,-46,0,-22,75,171,0,-28,0,6,27,6,0,-6,0,19,39,-75,0,79,-40,-28,-132,24,0,267,-1,30,-111,-252,-55,5,0,-198,-3,46,0,-81,49,-42,-32,528,0,-150,0,656,114,26,0,-433,-145,150,-63,-514,0,160,0

%N G.f. A(x) satisfies: Sum_{n>=0} A(x^n)^n = x.

%H Paul D. Hanna, <a href="/A326559/b326559.txt">Table of n, a(n) for n = 1..1300</a>

%F a(p^2) = -1 for prime p.

%e G.f.: A(x) = x - x^4 - x^9 + 2*x^10 - 2*x^16 + 3*x^18 + 2*x^20 - 4*x^22 - x^25 - 2*x^26 - 3*x^27 + 8*x^28 + 3*x^33 + 4*x^34 - 7*x^36 - 2*x^38 - 9*x^40 - 10*x^42 + 6*x^44 + 12*x^45 + 8*x^46 + 8*x^48 - x^49 - 4*x^50 + 3*x^51 - 2*x^52 + 16*x^54 - 10*x^55 - 6*x^56 - 3*x^57 - 14*x^58 - 21*x^60 - 14*x^62 - 24*x^63 + 50*x^64 + 5*x^65 - 3*x^66 + 4*x^68 + 6*x^69 - x^70 + 28*x^72 + 48*x^74 + 12*x^75 - 68*x^76 - 17*x^78 - 20*x^80 + 3*x^81 + ...

%e such that

%e x = A(x) + A(x^2)^2 + A(x^3)^3 + A(x^4)^4 + A(x^5)^5 + A(x^6)^6 + A(x^7)^7 + ...

%e TERMS AT SQUARE POSITIONS.

%e The terms a(n^2), for n >= 1 , begin:

%e [1, -1, -1, -2, -1, -7, -1, 50, 3, 171, -1, -433, -1, -1819, -493, -14972, -1, 60377, -1, -184172, -76687, 222858, -1, -374277, 208165, -4443338, 5349603, 75176806, -1, -43902199, -1, -1265783512, 691705812, 1753345623, -1500409643, -5035835610, ...].

%o (PARI) {a(n) = my(A=[1]); for(i=1,n,

%o A=concat(A,0); A[#A] = -polcoeff( sum(m=1,sqrtint(#A)+1, subst(x*Ser(A),x,x^m)^m ),#A););A[n]}

%o for(n=1,121,print1(a(n),", "))

%K sign

%O 1,10

%A _Paul D. Hanna_, Jul 14 2019

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)