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 A304399 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^4) / A(x) = 0 for n>0. 2

%I

%S 1,16,2200,1809920,4241345876,20919209023760,185887334702902784,

%T 2699985099706935115520,59877289873410663776378876,

%U 1926339929784486079047963326480,86370374435881318779333300624751016,5225229347181019896500110654738959018752,415299644168495653846091996394573044842672676

%N G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^4) / A(x) = 0 for n>0.

%e G.f.: A(x) = 1 + 16*x + 2200*x^2 + 1809920*x^3 + 4241345876*x^4 + 20919209023760*x^5 + 185887334702902784*x^6 + 2699985099706935115520*x^7 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in (1+x)^((n+1)^4)/A(x) begins:

%e n=0: [1, -15, -1960, -1745560, -4181956116, -20781289862564, ...;

%e n=1: [1, 0, -2080, -1776080, -4208350776, -20844203397376, ...;

%e n=2: [1, 65, 0, -1867600, -4327445336, -21121523038728, ...;

%e n=3: [1, 240, 26600, 0, -4559454036, -21903515092368, ...;

%e n=4: [1, 609, 183056, 34416384, 0, -23127137438064, ...;

%e n=5: [1, 1280, 816480, 344268080, 103140231304, 0, ...;

%e n=6: [1, 2385, 2840840, 2251489240, 1330416079284, 599753730572516, 0, ...; ...

%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)^4)/A(x) = 0 for n>0.

%e RELATED SERIES.

%e 1 - 1/A(x) = 16*x + 1944*x^2 + 1743616*x^3 + 4180212500*x^4 + 20777109650064*x^5 + 185199596154767936*x^6 + 2693946371100901126144*x^7 + ...

%e The logarithmic derivative of the g.f. A(x) begins

%e A'(x)/A(x) = 16 + 4144*x + 5328256*x^2 + 16842055888*x^3 + 104239488218896*x^4 + 1113257196684170944*x^5 + 18878740287619671915136*x^6 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m^4)/Ser(A) )[m] ); A[n+1]}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A304193, A304398.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 14 2018

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Last modified January 28 15:20 EST 2022. Contains 350657 sequences. (Running on oeis4.)