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A245493 a(n) = n! * [x^n] (exp(x)+x^2/2!)^n. 3
1, 1, 6, 45, 508, 7225, 126306, 2606065, 62075952, 1675774089, 50565938050, 1686510607111, 61609858744248, 2446470026497705, 104922088624078194, 4833250468667819325, 238004208840601580416, 12476420334546637657489, 693675026024580055139778 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

In general, if a(n) = n! * [x^n] (exp(x) + x^k/k!)^n, k>=1, then limit n-> infinity (a(n)/n!)^(1/n) = ((1-k*r)/(1-r))^(k-1) / (r*k!), where r is the root of the equation exp((k*r-1)/(1-r)) = r*k! * (1-r)^(k-1) / (1-k*r)^k.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

a(n) ~ c * d^n * n^n / exp(n), where d = (1-2*r)/(2*r*(1-r)) = 3.177499696443893762475339445134038..., where r = 0.13317988718414524112... is the root of the equation exp((2*r-1)/(1-r)) = 2*r*(1-r)/(1-2*r)^2, and c = 1.061620103934913384222610538939... .

MATHEMATICA

Table[n!*SeriesCoefficient[(E^x + x^2/2)^n, {x, 0, n}], {n, 0, 20}]

With[{k=2}, Flatten[{1, Table[Sum[Binomial[n, j]*Binomial[n, k*j]*(n-j)^(n-k*j)*(k*j)!/(k!)^j, {j, 0, n/k}], {n, 1, 20}]}]]

CROSSREFS

Cf. A245406, A245405, A245496.

Sequence in context: A186925 A294642 A109516 * A078865 A160492 A273091

Adjacent sequences:  A245490 A245491 A245492 * A245494 A245495 A245496

KEYWORD

nonn,easy

AUTHOR

Vaclav Kotesovec, Jul 24 2014

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.