A277380 -id:A277380 - cp's OEIS frontend

A277381 a(n) = exp(-1) * Sum_{k>=0} H_n(k)/k!, where H_n(x) is n-th Hermite polynomial.

1, 2, 6, 28, 156, 984, 7112, 57488, 508688, 4887840, 50639200, 561416640, 6621963712, 82719611264, 1089925195904, 15094182365440, 219035852333312, 3321573144437248, 52511336993334784, 863594855161338880, 14746060593172585472, 260971042697438779392

Offset: 0

Vladimir Reshetnikov, Oct 11 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Hermite Polynomial
Wikipedia, Hermite polynomials

Cf. A277380, A316778.

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(2*x) -x^2 -1))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 12 2018

g := exp(exp(2*x)-x^2-1): ser := series(g,x,32):
seq(n!*coeff(ser,x,n), n=0..21); # Peter Luschny, Oct 19 2016

Table[Sum[HermiteH[n, k]/k!, {k, 0, Infinity}]/E, {n, 0, 20}] (* or *)
Table[SeriesCoefficient[Exp[Exp[2 x] - x^2 - 1], {x, 0, n}] n!, {n, 0, 20}]
With[{nmax = 50}, CoefficientList[Series[Exp[Exp[2*x] - x^2 - 1], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 12 2018 *)
Table[Sum[Binomial[n, k] * 2^k * BellB[k] * HermiteH[n-k, 0], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 13 2018 *)

x='x+O('x^50); Vec(serlaplace(exp(exp(2*x)-x^2-1))) \\ G. C. Greubel, Jul 12 2018

E.g.f.: exp(exp(2*x)-x^2-1).

a(n) ~ 2^n * n^n * exp(n/LambertW(n) - LambertW(n)^2 / 4 - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^n). - Vaclav Kotesovec, Jun 29 2022

A316778 a(n) = exp(-1/2) * Sum_{k>=0} H_n(k) / (k!*2^k), where H_n(x) is n-th Hermite polynomial.

Original entry on oeis.org

1, 1, 1, 5, 25, 97, 489, 3285, 22481, 160737, 1293041, 11348933, 105136937, 1033279873, 10808289561, 119401994709, 1385242479137, 16846680046657, 214333419288161, 2844927602028549, 39305588104667321, 564208058072724257, 8400178767847987401, 129509650839484638037

Offset: 0

Vaclav Kotesovec, Jul 13 2018

nonn

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + q*x^2 + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) + q*LambertW(n/m)^2 / b^2 - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - Vaclav Kotesovec, Jun 29 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A277380, A277381.

Table[Exp[-1/2]*Sum[HermiteH[n, k]/k!/2^k, {k, 0, Infinity}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[Exp[2*x]/2 - x^2 - 1/2], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, 1/2] * HermiteH[n-k, 0], {k, 0, n}], {n, 0, 20}]

E.g.f.: exp(exp(2*x)/2 - x^2 - 1/2).

a(n) ~ 2^n * n^n * exp(n/LambertW(2*n) - LambertW(2*n)^2 / 4 - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^n). - Vaclav Kotesovec, Jun 29 2022

cp's OEIS Frontend

A277381 a(n) = exp(-1) * Sum_{k>=0} H_n(k)/k!, where H_n(x) is n-th Hermite polynomial.

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