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

1, 2, 10, 104, 1772, 42408, 1303504, 48736000, 2139552016, 107629121888, 6094743943584, 383305860004992, 26491391713168640, 1994924925169038848, 162537118868301414912, 14243360542620058589184, 1335710880923054761115904, 133461369304858515494530560, 14154134380237986764584033792, 1587931951984022880659170662400

Offset

0,2

Comments

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

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

Example

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 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n*(n+1)*x) / A(x) begins:
n=0: [1, -2, -12, -432, -32640, -4176000, -804504960, -216834831360, ...];
n=1: [1, 0, -16, -520, -36432, -4520768, -856647680, -228458074752, ...];
n=2: [1, 4, 0, -648, -46032, -5341824, -974612736, -254049782400, ...];
n=3: [1, 10, 84, 0, -56832, -6922368, -1194341760, -299397745152, ...];
n=4: [1, 18, 308, 4448, 0, -8528000, -1573784960, -376524725760, ...];
n=5: [1, 28, 768, 20088, 444720, 0, -1938504960, -502258872960, ...];
n=6: [1, 40, 1584, 61560, 2286768, 72032832, 0, -618983309952, ...];
n=7: [1, 54, 2900, 154352, 8074368, 404450176, 17201640064, 0, ...];
n=8: [1, 70, 4884, 339120, 23357568, 1583068032, 102886277760, 5682964174848, 0, ...]; ...
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.
LOGARITHMIC DERIVATIVE.
The logarithmic derivative of A(x) yields the o.g.f. of A304317:
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 + ...

Prog

(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]}
for(n=0, 25, print1( a(n), ", "))

Crossrefs

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

Sequence in context: A346672 A154256 A027270 * A005799 A208730 A000595

Adjacent sequences: A304316 A304317 A304318 * A304320 A304321 A304322

Keyword

nonn

Author

Paul D. Hanna, May 11 2018

Status

approved