A302189 -id:A302189 - cp's OEIS frontend

A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).

1, 1, 6, 39, 352, 3965, 53556, 844123, 15204960, 308118105, 6937562980, 171826160231, 4642588564032, 135891789038629, 4283619809941668, 144674451274329075, 5211965027738046016, 199498704931954788785, 8085413817213212761668, 345895984008645703002559

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..396

Cf. A000290, A006153, A033453, A033462, A302189, A308862.

nmax = 19; CoefficientList[Series[1/(1 - x (1 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

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

E.g.f.: 1 / (1 - Sum_{k>=1} k^2*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).
a(n) ~ n! / (r^(n+1) * exp(r) * (1 + 3*r + r^2)), where r = A201941 = 0.44413022882396659058546632949098466707932096994213775695918... is the root of the equation exp(r)*r*(1 + r) = 1. - Vaclav Kotesovec, Jun 29 2019

A302191 Numerators of Hurwitz inverse of primes [2,3,5,7,...].

Original entry on oeis.org

1, -3, 1, -5, -7, 97, -403, 3795, 1683, -67403, 141662, -5744835, -710829, 124489961, -7187558877, 247099181979, -43618981401, -2710990422171, 16455095049450, -1725616801459565, 2828334020055989, 58332444583336295, -2708485501761494555

Offset: 0

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

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

			1/2, -3/4, 1, -5/8, -7/2, 97/4, -403/4, 3795/16, 1683/2, -67403/4, 141662, -5744835/8, -710829/2, 124489961/2, -7187558877/8, ...

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

N. J. A. Sloane, Maple programs for operations on Hurwitz sequences

Cf. A302192, A302193.

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

E.g.f. for A302191/A302192 is 1 / Sum_{n >= 0} prime(n+1)*x^n/n!.

A302194 Hurwitz inverse of [1 followed by primes], [1,2,3,5,7,...].

Original entry on oeis.org

1, -2, 5, -17, 79, -471, 3391, -28451, 272447, -2933807, 35102403, -462021525, 6634207777, -103200019093, 1728836723813, -31030630439249, 594094812208133, -12085090282079299, 260296103744105623, -5917885334682695549, 141625618336446419151

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..436
N. J. A. Sloane, Maple programs for operations on Hurwitz sequences

Cf. A302191, A302192.

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

E.g.f. = 1 / (1 + Sum_{n >= 1} prime(n)*x^n/n!).

A302870 Expansion of e.g.f. 1 / Sum_{n >= 0} (n+1)^3*x^n/n!.

Original entry on oeis.org

1, -8, 101, -1840, 44441, -1340696, 48530653, -2049479216, 98915010545, -5370730092136, 324012625790741, -21502216185516848, 1556657523678767881, -122085765970981019000, 10311495889448094131981, -933128678308256836233136, 90072066063382006331898593

Offset: 0

Seiichi Manyama, Apr 15 2018

sign

From Vaclav Kotesovec, Apr 15 2018: (Start)
In general, for m>=0, Sum_{k>=0} (k+1)^m * x^k / k! = exp(x) * Sum_{j = 1..m+1} Stirling2(m+1, j) * x^(j-1).
If m tends to infinity, then the real root of the equation Sum_{j = 1..m+1} Stirling2(m + 1, j) * x^(j-1) = 0, with a minimal absolute value, tends to -1/2^m.
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..345

Cf. A302189.

nmax = 20; CoefficientList[Series[1/(E^x*((1 + 7*x + 6*x^2 + x^3))), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2018 *)

From Vaclav Kotesovec, Apr 15 2018: (Start)
E.g.f: exp(-x)/(1 + 7*x + 6*x^2 + x^3).
a(n) ~ (-1)^n * n! * (371 + 414*r + 74*r^2) * exp(-r) * (7 + 6*r + r^2)^n / 257, where r = -0.16575681568607828288437387419... is the real root of the equation 1 + 7*r + 6*r^2 + r^3 = 0.
(End)

A302197 Hurwitz logarithm of Catalan numbers [1,1,2,5,14,...].

Original entry on oeis.org

0, 1, 1, 1, 0, -4, -10, 15, 210, 504, -3528, -34440, -36960, 1512720, 11763180, -24549525, -1118467350, -6466860400, 62185563440, 1297024576848, 3903558763104, -149417396724960, -2150022118411440, 3233834859735480, 449839942314082320

Offset: 0

N. J. A. Sloane and William F. Keigher, Apr 14 2018

sign

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

V. E. Adler and A. B. Shabat, Volterra chain and Catalan numbers, arXiv:1810.13198 [nlin.SI], 2018.
Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885 .

Cf. A000108, A302189.

Cf. A005043.

# first load Maple commands for Hurwitz operations from link in A302189.
s:=[seq(binomial(2*n,n)/(n+1),n=0..30)];
Hlog(s);

nmax = 30; CoefficientList[Series[2*x + Log[BesselI[0, 2*x] - BesselI[1, 2*x]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 26 2023 *)

A = PowerSeriesRing(QQ, 'x')
f = A([catalan_number(i) for i in range(30)]).ogf_to_egf().log()
print(list(f.egf_to_ogf()))
# F. Chapoton, Apr 11 2020

E.g.f. is log of e.g.f. for Catalan numbers.

E.g.f. is also the log of e^x times the e.g.f. of A005043. - Tom Copeland, Jun 26 2023

A302195 Hurwitz inverse of triangular numbers [1,3,6,10,15,...].

Original entry on oeis.org

1, -3, 12, -64, 441, -3771, 38638, -461742, 6306009, -96885451, 1653938616, -31057949748, 636230845297, -14119481897379, 337448486204586, -8640908986912786, 236015269236658833, -6849355531826261427, 210466462952536609924

Offset: 0

N. J. A. Sloane and William F. Keigher, Apr 14 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..400

Cf. A000217, A302189.

# first load Maple commands for Hurwitz operations from link in A302189.
s:=[seq(n*(n+1)/2,n=1..64)];
Hinv(s);

nmax = 20; CoefficientList[Series[1/(E^x*(1 + 2*x + x^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 26 2018 *)

E.g.f. = 1 / Sum_{n >= 0} ((n+1)*(n+2)/2)*x^n/n!.
From Vaclav Kotesovec, Apr 26 2018: (Start)
E.g.f: exp(-x) / (1 + 2*x + x^2/2).
a(n) ~ (-1)^n * n! * exp(2 - sqrt(2)) * (1 + 1/sqrt(2))^(n+1) / sqrt(2).
(End)

A302199 Hurwitz inverse of partition numbers A000041.

Original entry on oeis.org

1, -1, 0, 3, -5, -17, 103, 55, -2680, 6720, 82446, -642698, -2087303, 53641331, -96015983, -4454066000, 35131380473, 323923309109, -6776856484915, -3620043398324, 1159030195119059, -7865002945782432, -175052008152354596, 3163635176513031787

Offset: 0

N. J. A. Sloane and William F. Keigher, Apr 14 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..482

Cf. A000041, A302189.

# first load Maple commands for Hurwitz operations from link in A302189.
with(combinat);
s:=[seq(numbpart,n=0..40)];
Hinv(s);

E.g.f. = 1 / Sum_{n >= 0} partition(n)*x^n/n!.

A302190 Hurwitz logarithm of natural numbers 1,2,3,4,5,...

Original entry on oeis.org

0, 2, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800, -39916800, 479001600, -6227020800, 87178291200, -1307674368000, 20922789888000, -355687428096000, 6402373705728000, -121645100408832000, 2432902008176640000, -51090942171709440000

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. See Ex. 2.16.
N. J. A. Sloane, Maple programs for operations on Hurwitz sequences

Cf. A133942.

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

A = PowerSeriesRing(QQ, 'x')
f = A(list(range(1,30))).ogf_to_egf().log()
print(list(f.egf_to_ogf()))
# F. Chapoton, Apr 11 2020

E.g.f. is log of Sum_{n >= 0} (n+1)*x^n/n!.

A335577 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).

Original entry on oeis.org

1, -1, -2, 9, 32, -285, -1236, 18725, 86176, -2087001, -9204580, 351964569, 1336442304, -83422970917, -231889447076, 26389118293005, 35917342192064, -10722110983670193, 5028963509133756, 5432569724760331841, -14852185163192897120, -3352369390318855889661

Offset: 0

Ilya Gutkovskiy, Jan 26 2021

sign

Cf. A302189, A302397, A308861, A316087, A335578.

a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
nmax = 21; CoefficientList[Series[1/(1 + Exp[x] x (1 + x)), {x, 0, nmax}], x] Range[0, nmax]!

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

E.g.f.: 1 / (1 + Sum_{k>=1} k^2 * x^k / k!).

A302196 Hurwitz logarithm of triangular numbers [1,3,6,10,15,...].

Original entry on oeis.org

0, 3, -3, 10, -51, 348, -2970, 30420, -363510, 4964400, -76272840, 1302058800, -24450287400, 500871016800, -11115524019600, 265655410020000, -6802532278542000, 185802383710944000, -5392136656290384000, 165689154918679392000, -5374132518684161232000, 183484361312817364800000

Offset: 0

N. J. A. Sloane and William F. Keigher, Apr 14 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.

Cf. A000217, A302189.

# first load Maple commands for Hurwitz operations from link in A302189.
s:=[seq(n*(n+1)/2,n=1..64)];
Hlog(s);

A = PowerSeriesRing(QQ, 'x')
f = A([binomial(i+2,2) for i in range(30)]).ogf_to_egf().log()
print(list(f.egf_to_ogf()))
#F. Chapoton, Apr 11 2020

E.g.f. is log of Sum_{n >= 0} ((n+1)*(n+2)/2)*x^n/n!.

cp's OEIS Frontend

A308861 Expansion of e.g.f. 1/(1 - x*(1 + x)*exp(x)).

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A302191 Numerators of Hurwitz inverse of primes [2,3,5,7,...].

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A302194 Hurwitz inverse of [1 followed by primes], [1,2,3,5,7,...].

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A302870 Expansion of e.g.f. 1 / Sum_{n >= 0} (n+1)^3*x^n/n!.

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A302197 Hurwitz logarithm of Catalan numbers [1,1,2,5,14,...].

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A302195 Hurwitz inverse of triangular numbers [1,3,6,10,15,...].

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A302199 Hurwitz inverse of partition numbers A000041.

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A302190 Hurwitz logarithm of natural numbers 1,2,3,4,5,...

A308861 Expansion of e.g.f. 1/(1 - x(1 + x)exp(x)).