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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: 0 = [x^n] Sum_{m=0..n+1} A(x)^(m^2) / m!, for n > 1.
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%I #5 Dec 30 2021 01:57:31

%S -1,1,6,76,74,26058,-7794632,-14388774920,213380711849598,

%T 28487519401892735386,-38029127603565193657832846,

%U -558442679082054075230555562505128,98405860538000983882824515617294391388622,225427456282611271896161167867656072505435638401710

%N G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: 0 = [x^n] Sum_{m=0..n+1} A(x)^(m^2) / m!, for n > 1.

%C This sequence is conjectured to consist entirely of integers.

%e G.f.: A(x) = -1 + x + 6*x^2 + 76*x^3 + 74*x^4 + 26058*x^5 - 7794632*x^6 - 14388774920*x^7 + 213380711849598*x^8 + 28487519401892735386*x^9 + ...

%e The table of coefficients of x^k in A(x)^(n^2), k >= 0, begins:

%e n=1: [-1, 1, 6, 76, 74, 26058, -7794632, ...];

%e n=2: [1, -4, -18, -236, 761, -99192, 31519032, ...];

%e n=3: [-1, 9, 18, 336, -4716, 221688, -72062976, ...];

%e n=4: [1, -16, 24, -336, 13116, -438576, 130813640, ...];

%e n=5: [-1, 25, -150, 600, -25800, 855780, -210360800, ...];

%e n=6: [1, -36, 414, -2316, 46161, -1632600, 316244424, ...];

%e n=7: [-1, 49, -882, 8036, -97706, 3042606, -459537288, ...]; ...

%e in which, by definition, the following sums along the columns equal zero:

%e 0 = (6)/1! + (-18)/2! + (18)/3! ;

%e 0 = (76)/1! + (-236)/2! + (336)/3! + (-336)/4! ;

%e 0 = (74)/1! + (761)/2! + (-4716)/3! + (13116)/4! + (-25800)/5! ;

%e 0 = (26058)/1! + (-99192)/2! + (221688)/3! + (-438576)/4! + (855780)/5! + (-1632600)/6! ;

%e 0 = (-7794632)/1! + (31519032)/2! + (-72062976)/3! + (130813640)/4! + (-210360800)/5! + (316244424)/6! + (-459537288)/7! ; ...

%e One may continue the above pattern to determine all the terms of this sequence.

%o (PARI) {a(n) = my(A=[-1,1,0]); for(i=1,n, A=concat(A,0);

%o A[#A-1] = (-1)^(#A-1) * (#A-2)! * polcoeff( sum(m=0,#A-1, Ser(A)^(m^2) / m!) ,#A-2) );A[n+1]}

%o for(n=0,16,print1(a(n),", "))

%Y Cf. A350410, A350412.

%K sign

%O 0,3

%A _Paul D. Hanna_, Dec 29 2021