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%I A350412 #9 Dec 30 2021 01:57:38
%S A350412 1,12,830,1867901,251714851830,3696867014099083814,
%T A350412 8468768030682252554158546818,
%U A350412 4074040763513889480730186336041282515610,533088609878310228401628064629048305794583364205468626,23856105278172150534783174192628043609641782797360374708050534816035626
%N A350412 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 A350412 This sequence is conjectured to consist entirely of integers.
%e A350412 G.f.: A(x) = x + 12*x^2 + 830*x^3 + 1867901*x^4 + 251714851830*x^5 + 3696867014099083814*x^6 + 8468768030682252554158546818*x^7 + ...
%e A350412 The table of coefficients of x^k in (A(x) - 1)^(n^2), k >= 0, begins:
%e A350412 n=1: [-1, 1, 12, 830, 1867901, 251714851830, ...];
%e A350412 n=2: [1, -4, -42, -3180, -7460923, -1006836884628, ...];
%e A350412 n=3: [-1, 9, 72, 6690, 16749063, 2265298700004, ...];
%e A350412 n=4: [1, -16, -72, -10960, -29688276, -4026988495968, ...];
%e A350412 n=5: [-1, 25, 0, 15850, 46226475, 6291750745680, ...];
%e A350412 n=6: [1, -36, 198, -21900, -66306051, -9059386973652, ...];
%e A350412 n=7: [-1, 49, -588, 30870, 89857033, 12329656582362, ...];
%e A350412 n=8: [1, -64, 1248, -46400, -116773328, -16102277850240, ...];
%e A350412 n=9: [-1, 81, -2268, 74790, 146862801, 20376929526066, ...];
%e A350412 n=10: [1, -100, 3750, -125900, -179760275, -25153253915220, ...]; ...
%e A350412 in which, by definition, the following sums along the columns equal zero:
%e A350412 0 = (12)/1! + (-42)/2! + (72)/3! + (-72)/4! ;
%e A350412 0 = (830)/1! + (-3180)/2! + (6690)/3! + (-10960)/4! + (15850)/5! + (-21900)/6! ;
%e A350412 0 = (1867901)/1! + (-7460923)/2! + (16749063)/3! + (-29688276)/4! + (46226475)/5! + (-66306051)/6! + (89857033)/7! + (-116773328)/8! ;
%e A350412 0 = (251714851830)/1! + (-1006836884628)/2! + (2265298700004)/3! + (-4026988495968)/4! + (6291750745680)/5! + (-9059386973652)/6! + (12329656582362)/7! + (-16102277850240)/8! + (20376929526066)/9! + (-25153253915220)/10! ; ...
%e A350412 One may continue the above pattern to determine all the terms of this sequence.
%o A350412 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A350412 A[#A] = (2*#A-1)! * polcoeff( sum(m=0,2*#A, (-1 + x*Ser(A))^(m^2) / m!) ,#A) );A[n]}
%o A350412 for(n=1,15,print1(a(n),", "))
%Y A350412 Cf. A350410, A350411.
%K A350412 nonn
%O A350412 1,2
%A A350412 _Paul D. Hanna_, Dec 29 2021

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