A302870 -id:A302870 - cp's OEIS frontend

A302189 Hurwitz inverse of squares [1,4,9,16,...].

1, -4, 23, -184, 1933, -25316, 397699, -7288408, 152650649, -3596802148, 94165506031, -2711813462744, 85195437862693, -2899579176456964, 106276755720182363, -4173542380352243896, 174823612884063939889, -7780800729631450594628

Offset: 0

N. J. A. Sloane and William F. Keigher, Apr 12 2018

sign

In the ring of Hurwitz sequences all members have offset 0.

Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885

Seiichi Manyama, Table of n, a(n) for n = 0..387
N. J. A. Sloane, Maple programs for operations on Hurwitz sequences

Cf. A302870.

# first load Maple commands for Hurwitz operations from link
s:=[seq(n^2,n=1..64)];
Hinv(s);

nmax = 20; CoefficientList[Series[1/(E^x*(1 + 3*x + x^2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2018 *)

E.g.f. = 1 / Sum_{n >= 0} (n+1)^2*x^n/n!.
From Vaclav Kotesovec, Apr 15 2018: (Start)
E.g.f: exp(-x)/(1 + 3*x + x^2).
a(n) ~ (-1)^n * n! * exp(1/phi^2) * phi^(2*n + 2) / sqrt(5), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio.
(End)

A308862 Expansion of e.g.f. 1/(1 - x(1 + 3x + x^2)*exp(x)).

Original entry on oeis.org

1, 1, 10, 81, 976, 14505, 258456, 5377897, 127852096, 3419620209, 101625743080, 3322169384721, 118475520287136, 4577175039397753, 190436902905933880, 8489222610046324665, 403657900923994965376, 20393319895130130117729, 1090902632352025316904648

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381

Cf. A000578, A006153, A144109, A279358, A302870, A308861.

nmax = 18; CoefficientList[Series[1/(1 - x (1 + 3 x + x^2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1 - x*(1 + 3*x + x^2)*exp(x)))) \\ Michel Marcus, Mar 10 2022

E.g.f.: 1 / (1 - Sum_{k>=1} k^3*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).
a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 7*r + 6*r^2 + r^3)), where r = 0.33649177041401456061485914122406146158245451810028937972189... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - Vaclav Kotesovec, Jun 29 2019

A335578 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).

Original entry on oeis.org

1, -1, -6, 15, 272, -745, -29976, 61271, 6065856, -2723697, -1941455080, -3989345041, 897021218400, 4964061925511, -562221881675832, -5689641396555705, 456732442022509184, 7321841133968133023, -464200472167634521800, -10961686347887871324289, 573373115861405030522400

Offset: 0

Ilya Gutkovskiy, Jan 26 2021

sign

Cf. A302397, A302870, A308862, A316088, A335577.

a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[1/(1 + Exp[x] x (1 + 3 x + x^2)), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 + exp(x) * x * (1 + 3*x + x^2)).

E.g.f.: 1 / (1 + Sum_{k>=1} k^3 * x^k / k!).

cp's OEIS Frontend

A302189 Hurwitz inverse of squares [1,4,9,16,...].

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A308862 Expansion of e.g.f. 1/(1 - x(1 + 3x + x^2)*exp(x)).

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A335578 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).

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cp's OEIS Frontend

A302189 Hurwitz inverse of squares [1,4,9,16,...].

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A308862 Expansion of e.g.f. 1/(1 - x*(1 + 3*x + x^2)*exp(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A335578 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A308862 Expansion of e.g.f. 1/(1 - x(1 + 3x + x^2)*exp(x)).