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A304319 O.g.f. A(x) satisfies: [x^n] exp( n*(n+1) * x ) / A(x) = 0 for n>0. 7

%I #18 Sep 04 2020 10:42:02

%S 1,2,10,104,1772,42408,1303504,48736000,2139552016,107629121888,

%T 6094743943584,383305860004992,26491391713168640,1994924925169038848,

%U 162537118868301414912,14243360542620058589184,1335710880923054761115904,133461369304858515494530560,14154134380237986764584033792,1587931951984022880659170662400

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

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

%C Note that if [x^n] exp( (n+1)*(n+2)*x ) / G(x) = 0 then G(x) does not consist entirely of integer coefficients.

%H Paul D. Hanna, <a href="/A304319/b304319.txt">Table of n, a(n) for n = 0..400</a>

%F a(n) ~ sqrt(1-c) * 2^(2*n) * 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 + 10*x^2 + 104*x^3 + 1772*x^4 + 42408*x^5 + 1303504*x^6 + 48736000*x^7 + 2139552016*x^8 + 107629121888*x^9 + 6094743943584*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, -2, -12, -432, -32640, -4176000, -804504960, -216834831360, ...];

%e n=1: [1, 0, -16, -520, -36432, -4520768, -856647680, -228458074752, ...];

%e n=2: [1, 4, 0, -648, -46032, -5341824, -974612736, -254049782400, ...];

%e n=3: [1, 10, 84, 0, -56832, -6922368, -1194341760, -299397745152, ...];

%e n=4: [1, 18, 308, 4448, 0, -8528000, -1573784960, -376524725760, ...];

%e n=5: [1, 28, 768, 20088, 444720, 0, -1938504960, -502258872960, ...];

%e n=6: [1, 40, 1584, 61560, 2286768, 72032832, 0, -618983309952, ...];

%e n=7: [1, 54, 2900, 154352, 8074368, 404450176, 17201640064, 0, ...];

%e n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 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 A304317:

%e A'(x)/A(x) = 2 + 16*x + 260*x^2 + 6200*x^3 + 191832*x^4 + 7235152*x^5 + 320372320*x^6 + 16243028896*x^7 + 926219213216*x^8 + 58608051937536*x^9 + 4072302306624576*x^10 + ...+ A304317(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*(m-1) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}

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

%Y Cf. A304317, A304318, A304320, A304863, A304864, A304865.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 11 2018

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)