%I #9 Dec 30 2021 01:57:38

%S 1,12,830,1867901,251714851830,3696867014099083814,

%T 8468768030682252554158546818,

%U 4074040763513889480730186336041282515610,533088609878310228401628064629048305794583364205468626,23856105278172150534783174192628043609641782797360374708050534816035626

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

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

%e G.f.: A(x) = x + 12*x^2 + 830*x^3 + 1867901*x^4 + 251714851830*x^5 + 3696867014099083814*x^6 + 8468768030682252554158546818*x^7 + ...

%e The table of coefficients of x^k in (A(x) - 1)^(n^2), k >= 0, begins:

%e n=1: [-1, 1, 12, 830, 1867901, 251714851830, ...];

%e n=2: [1, -4, -42, -3180, -7460923, -1006836884628, ...];

%e n=3: [-1, 9, 72, 6690, 16749063, 2265298700004, ...];

%e n=4: [1, -16, -72, -10960, -29688276, -4026988495968, ...];

%e n=5: [-1, 25, 0, 15850, 46226475, 6291750745680, ...];

%e n=6: [1, -36, 198, -21900, -66306051, -9059386973652, ...];

%e n=7: [-1, 49, -588, 30870, 89857033, 12329656582362, ...];

%e n=8: [1, -64, 1248, -46400, -116773328, -16102277850240, ...];

%e n=9: [-1, 81, -2268, 74790, 146862801, 20376929526066, ...];

%e n=10: [1, -100, 3750, -125900, -179760275, -25153253915220, ...]; ...

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

%e 0 = (12)/1! + (-42)/2! + (72)/3! + (-72)/4! ;

%e 0 = (830)/1! + (-3180)/2! + (6690)/3! + (-10960)/4! + (15850)/5! + (-21900)/6! ;

%e 0 = (1867901)/1! + (-7460923)/2! + (16749063)/3! + (-29688276)/4! + (46226475)/5! + (-66306051)/6! + (89857033)/7! + (-116773328)/8! ;

%e 0 = (251714851830)/1! + (-1006836884628)/2! + (2265298700004)/3! + (-4026988495968)/4! + (6291750745680)/5! + (-9059386973652)/6! + (12329656582362)/7! + (-16102277850240)/8! + (20376929526066)/9! + (-25153253915220)/10! ; ...

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

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

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

%o for(n=1,15,print1(a(n),", "))

%Y Cf. A350410, A350411.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 29 2021