A277381 -id:A277381 - cp's OEIS frontend

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

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

A355337 Expansion of e.g.f.: exp(exp(x) + x^2 - 1).

Original entry on oeis.org

1, 1, 4, 11, 51, 212, 1133, 6001, 36508, 228435, 1559575, 11079180, 83753497, 659858617, 5459331036, 46980355355, 421272977267, 3917446787884, 37766791690501, 376447420971545, 3875957531387172, 41149332371734371, 449984429580538407, 5061923434006018612, 58517321729774406129

Offset: 0

Vaclav Kotesovec, Jun 29 2022

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)).

Number of ways the roots of a polynomial with real coefficients and degree n can be configured regarding multiplicity and complexity. By configuration we mean for example a product of the form (x-b)*(x-c)*...; the roots of a polynomial do not imply any order, but the parameters which define roots may be labeled. In the case of a conjugate complex pair, we will distinguish between positive and negative imaginary part. For details see the example for a(4) in the "LINKS" section. - Thomas Scheuerle, Jun 01 2024

Vaclav Kotesovec, Table of n, a(n) for n = 0..568
Thomas Scheuerle, An example for a(4) = 51 which explains the relation to polynomial roots.

Cf. A000110, A277381, A316778, A355338.

nmax = 25; CoefficientList[Series[Exp[Exp[x] + x^2 - 1], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x) + x^2 - 1))) \\ Michel Marcus, Jun 29 2022

a(n) = sum(k=0, floor(n/2), sum(m=0, n-(k*2), stirling(n-(k*2), m, 2))*(2*k)!/k!*binomial(n, n-(k*2))) \\ Thomas Scheuerle, Jun 01 2024

a(n) ~ n^n * exp(n/LambertW(n) + LambertW(n)^2 - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^n).
a(n) ~ Bell(n) * exp(LambertW(n)^2).
a(0) = a(1) = 1; a(n) = 2 * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 29 2022
a(n) = Sum_{k=0..floor(n/2)} A000110(n - k*2)*(2*k)!/k!*binomial(n, n - (k*2)). - Thomas Scheuerle, Jun 01 2024

A277380 a(n) = Sum_{k>=1} H_n(k-1)/2^k, where H_n(x) is n-th Hermite polynomial.

Original entry on oeis.org

1, 2, 10, 92, 1068, 15352, 265752, 5368400, 123919248, 3217983008, 92851377312, 2947037232064, 102040223376576, 3827536020146048, 154615082607931776, 6691872388083371264, 308938595472492867840, 15153942107317778727424, 787050616613300039649792

Offset: 0

Vladimir Reshetnikov, Oct 11 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..380
Eric Weisstein's World of Mathematics, Hermite Polynomial
Wikipedia, Hermite polynomials

Cf. A277381, A316778.

Table[Sum[HermiteH[n, k - 1]/2^k, {k, 1, Infinity}], {n, 0, 20}]

for(n=0,40, print1(if(n==0,1, ceil(sum(k=1, 15*n, polhermite(n, k-1)/2^k))), ", ")) \\ G. C. Greubel, Jul 13 2018

nmax = 40; p = floor(2*log(nmax!*(2/log(2))^nmax)/log(10)); default(realprecision, p); a(n) = round(suminf(k=1, polhermite(n, k-1)/2^k));
for(n=0, nmax, print1(a(n), ", ")); \\ Michel Marcus and Vaclav Kotesovec, Jul 13 2018

a(n) ~ c * 2^n * n! / (log(2))^n, where c = 1/(2 * log(2) * exp(log(2)^2/4)) = 0.639705404891769467944095575437073306645289161842121830191257596548619914238... - Vaclav Kotesovec, Jul 13 2018, updated Apr 21 2024

A355338 Expansion of e.g.f.: exp(exp(x) - x^2 - 1).

Original entry on oeis.org

1, 1, 0, -1, 3, 12, -7, -47, 332, 1347, -2105, -4200, 135457, 474697, -900832, 4682135, 126196787, 439488524, 233313817, 19129265609, 239146712732, 1104038984091, 5891696027079, 89831511761320, 911995655018817, 6253185308181553, 54873149768926624, 653039078246798383

Offset: 0

Vaclav Kotesovec, Jun 29 2022

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..578

Cf. A000296, A277381, A316778, A355337.

nmax = 30; CoefficientList[Series[Exp[Exp[x] - x^2 - 1], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x) - x^2 - 1))) \\ Michel Marcus, Jun 29 2022

a(n) ~ n^n * exp(n/LambertW(n) - LambertW(n)^2 - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^n).
a(n) ~ Bell(n) / exp(LambertW(n)^2).
a(0) = a(1) = 1; a(n) = -2 * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 29 2022

cp's OEIS Frontend

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

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A355337 Expansion of e.g.f.: exp(exp(x) + x^2 - 1).

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A277380 a(n) = Sum_{k>=1} H_n(k-1)/2^k, where H_n(x) is n-th Hermite polynomial.

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A355338 Expansion of e.g.f.: exp(exp(x) - x^2 - 1).

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Mathematica

PARI

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