A304318 - OEIS

%I #14 Sep 04 2020 11:47:18

%S 1,0,2,24,436,10656,328112,12183456,529242224,26309617536,

%T 1472135847072,91526938123008,6258004268952064,466599240364076544,

%U 37672137946943244288,3274012281487011586560,304724394621209905647360,30239686358027369113804800,3187164738879981461171955200,355548230503664593634743375872

%N O.g.f. A(x) satisfies: [x^n] exp( n*(n-1) * x ) / A(x) = 0.

%C It is striking that the coefficients of o.g.f. A(x) consist entirely of integers.

%H Paul D. Hanna, <a href="/A304318/b304318.txt">Table of n, a(n) for n = 0..400</a>

%F a(n) ~ sqrt(1-c) * 2^(2*n - 1) * n^(n - 1/2) / (sqrt(Pi) * c^(n - 1/2) * (2-c)^n * exp(n)), where c = -A226775 = -LambertW(-2*exp(-2)). - _Vaclav Kotesovec_, Aug 31 2020

%e O.g.f.: A(x) = 1 + 2*x^2 + 24*x^3 + 436*x^4 + 10656*x^5 + 328112*x^6 + 12183456*x^7 + 529242224*x^8 + 26309617536*x^9 + 1472135847072*x^10 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp(n*(n-1)*x) / A(x) begins:

%e n=0: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];

%e n=1: [1, 0, -4, -144, -10368, -1267200, -234576000, -61085767680, ...];

%e n=2: [1, 2, 0, -160, -11600, -1376928, -250428416, -64479262720, ...];

%e n=3: [1, 6, 32, 0, -13392, -1630944, -286447104, -71981250048, ...];

%e n=4: [1, 12, 140, 1440, 0, -1916928, -351444096, -85338800640, ...];

%e n=5: [1, 20, 396, 7616, 128512, 0, -417488000, -107269127680, ...];

%e n=6: [1, 30, 896, 26496, 760752, 19101600, 0, -128348167680, ...];

%e n=7: [1, 42, 1760, 73440, 3034800, 121743072, 4260708864, 0, ...];

%e n=8: [1, 56, 3132, 174800, 9716608, 535021056, 28597069696, 1331047703552, 0, ...]; ...

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

%e LOGARITHMIC DERIVATIVE.

%e The logarithmic derivative of A(x) yields the o.g.f. of A304316:

%e A'(x)/A(x) = 4*x + 72*x^2 + 1736*x^3 + 53040*x^4 + 1961728*x^5 + 85062432*x^6 + 4225904800*x^7 + 236455369344*x^8 + 14705880874944*x^9 + 1005982098054912*x^10 + ... + A304316(n)*x^n + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(m-2) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}

%o for(n=0,25, print1( a(n),", "))

%Y Cf. A304316, A304319, A304320.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 11 2018