A302583 -id:A302583 - cp's OEIS frontend

A062024 a(n) = ((n+1)^n + (n-1)^n)/2.

1, 1, 5, 36, 353, 4400, 66637, 1188544, 24405761, 567108864, 14712104501, 421504185344, 13218256749601, 450353989316608, 16565151205544957, 654244800082329600, 27614800115689879553, 1240529732459024678912, 59095217374989483261925, 2975557672677668838178816

Offset: 0

Amarnath Murthy, Jun 02 2001

nonn

Let b(n) = A302583(n) = ((n+1)^n - (n-1)^n)/2 = 0, 1, 4, 28, 272, ... then lim_{n -> infinity} b(n)/a(n) = tanh(1) = 0.76159415... . - Thomas Ordowski, Dec 06 2012

Obviously, a(n) is always odd number for even n. - Altug Alkan, Sep 28 2015

			a(3) = (4^3 + 2^3)/2 = 36.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A302583.

List([0..20], n-> ((n+1)^n + (n-1)^n)/2); # G. C. Greubel, Jan 03 2020

[((n+1)^n + (n-1)^n)/2: n in [0..20]]; // Vincenzo Librandi, Sep 28 2015

A062024:=n->((n+1)^n + (n-1)^n)/2; seq(A062024(n), n=0..20); # Wesley Ivan Hurt, Nov 13 2013

a[n_]:=((n-1)^n+(n+1)^n)/2; a[Range[0, 20]] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010; modified by G. C. Greubel, Jan 03 2020 *)
Table[((n+1)^n + (n-1)^n)/2, {n,0,20}] (* Vincenzo Librandi, Sep 28 2015 *)

a(n) = { ((n + 1)^n + (n - 1)^n)/2 } \\ Harry J. Smith, Jul 29 2009

[((n+1)^n + (n-1)^n)/2 for n in (0..20)] # G. C. Greubel, Jan 03 2020

a(n) = n! * [x^n] exp(n*x)*cosh(x). - Ilya Gutkovskiy, Apr 10 2018

More terms from Larry Reeves (larryr(AT)acm.org) and Jason Earls, Jun 06 2001
Offset changed from 1 to 0 by Harry J. Smith, Jul 29 2009
a(18)-a(19) from Vincenzo Librandi, Sep 28 2015

A302587 a(n) = n! * [x^n] exp(nx)tanh(x).

Original entry on oeis.org

0, 1, 4, 25, 224, 2641, 38592, 671665, 13548544, 310580161, 7971353600, 226406902921, 7049219383296, 238722074157841, 8735529994928128, 343474252543881313, 14441163232204292096, 646510839624706118401, 30704150325602206089216, 1541807339347429264648441

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..386
N. J. A. Sloane, Transforms

Cf. A000182, A009832, A062024, A302583, A302584, A302585, A302586.

N:= 100: # to get a(0)..a(N)
T:= series(tanh(x),x,N+1):
C:= [seq(coeff(T,x,j),j=1..N)]:
seq(n! * add(C[i]*n^(n-i)/(n-i)!,i=1..n,2), n=0..N); # Robert Israel, Apr 10 2018

Table[n! SeriesCoefficient[Exp[n x] Tanh[x], {x, 0, n}], {n, 0, 19}]

a(n) = my(x='x+O('x^(n+1))); polcoeff(n!*exp(n*x)*tanh(x), n); \\ Michel Marcus, Apr 11 2018; corrected Jun 15 2022

a(n) ~ tanh(1) * n^n. - Vaclav Kotesovec, Jun 08 2019

A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

Original entry on oeis.org

1, 1, 5, 36, 357, 4500, 68857, 1239504, 25661545, 600655824, 15684383021, 452001644864, 14249852124365, 487836995500608, 18022519535240417, 714658089577017600, 30275849571771536977, 1364687729891761740032, 65213822241378992547925, 3293203845745202062590976

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A003701, A062024, A115415, A115416, A302583, A302585, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cos[x], {x, 0, n}], {n, 0, 19}]
Table[(2 I)^n EulerE[n, (1 - I n)/2], {n, 0, 19}]

a(n) ~ n^n / cos(1). - Vaclav Kotesovec, Jun 08 2019

A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

Original entry on oeis.org

1, 1, 3, 18, 165, 2000, 29855, 527632, 10762857, 248811264, 6428081979, 183537694208, 5739195739277, 195059957567488, 7159662639822615, 282252719348582400, 11894243092571825745, 533554809104057434112, 25384473065818477067123, 1276688324194885747474432

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A062024, A119880, A119881, A155585, A302583, A302584, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cosh[x], {x, 0, n}], {n, 0, 19}]
Table[2^n EulerE[n, (n + 1)/2], {n, 0, 19}]

a(n) ~ n^n / cosh(1). - Vaclav Kotesovec, Jun 08 2019

A302586 a(n) = n! * [x^n] exp(nx)tan(x).

Original entry on oeis.org

0, 1, 4, 29, 288, 3641, 55872, 1008349, 20923392, 490730641, 12836633600, 370512824285, 11697136754688, 400947361714121, 14829211483455488, 588633245015433437, 24960134277040177152, 1126038686507284428961, 53851620649898789830656, 2721385807644104827095965

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000182, A009739, A115415, A115416, A293020, A302583, A302584, A302585, A302587.

Table[n! SeriesCoefficient[Exp[n x] Tan[x], {x, 0, n}], {n, 0, 19}]
Table[I^(n + 1) 2^(n - 1) (EulerE[n, (-I/2) n] - EulerE[n, 1 - (I/2) n]), {n, 0, 19}]

a(n) ~ tan(1) * n^n. - Vaclav Kotesovec, Jun 08 2019

A302605 a(n) = n! * [x^n] exp(nx)arcsin(x).

Original entry on oeis.org

0, 1, 4, 28, 272, 3384, 51300, 917064, 18884672, 440168832, 11454902500, 329208395264, 10355322975120, 353851897861760, 13052503620917124, 516917167506777600, 21875427250996723968, 985164766018898243584, 47043119138733155306052, 2374168079889664129576960

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..384
N. J. A. Sloane, Transforms

Cf. A001818, A115416, A291482, A293191, A302583, A302584, A302585, A302586, A302587, A302606, A302608, A302609.

Table[n! SeriesCoefficient[Exp[n x] ArcSin[x], {x, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=1..n} binomial(n,k)*(k-2)!!^2*n^(n-k)*(1-(-1)^k)/2. - Fabian Pereyra, Oct 05 2024

A302609 a(n) = n! * [x^n] exp(nx)arctanh(x).

Original entry on oeis.org

0, 1, 4, 29, 288, 3649, 56160, 1017029, 21181440, 498682881, 13095232000, 379443829709, 12025239367680, 413761766695809, 15360425115176960, 611958601019294325, 26042588632355176448, 1179009749826940037889, 56579126414696034729984, 2868848293506101088635389

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..384
N. J. A. Sloane, Transforms

Cf. A010050, A291484, A293193, A302583, A302584, A302585, A302586, A302587, A302605, A302606, A302608.

Table[n! SeriesCoefficient[Exp[n x] ArcTanh[x], {x, 0, n}], {n, 0, 19}]
nmax = 20; CoefficientList[Series[Log[(1 - LambertW[-x])/(1 + LambertW[-x])] / (2*(1 + LambertW[-x])), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 09 2019 *)

E.g.f.: log((1 - LambertW(-x))/(1 + LambertW(-x))) / (2*(1 + LambertW(-x))). - Vaclav Kotesovec, Jun 09 2019
a(n) ~ log(n) * n^n / 4 * (1 + (gamma + 3*log(2))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 09 2019
a(n) = Sum_{k=1..n} binomial(n,k)*(k-1)!*n^(n-k)*(1-(-1)^k)/2. - Fabian Pereyra, Oct 05 2024

A302606 a(n) = n! * [x^n] exp(nx)arcsinh(x).

Original entry on oeis.org

0, 1, 4, 26, 240, 2884, 42660, 748544, 15185856, 349574544, 9000902500, 256293989984, 7996078704240, 271246034903232, 9939835626507332, 391303051339622400, 16469438021801262848, 737992773619777599744, 35077254665501330210628, 1762671472887447792620032

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A001818, A002866, A291483, A302583, A302584, A302585, A302586, A302587, A302605, A302608, A302609.

Table[n! SeriesCoefficient[Exp[n x] ArcSinh[x], {x, 0, n}], {n, 0, 19}]

a(n) ~ arcsinh(1) * n^n = log(1 + sqrt(2)) * n^n. - Vaclav Kotesovec, Jun 09 2019

a(n) = Sum_{k=1..n, k odd} (-1)^((k-1)/2)*binomial(n,k)*(k-2)!!^2*n^(n-k). - Fabian Pereyra, Oct 05 2024

A302608 a(n) = n! * [x^n] exp(nx)arctan(x).

Original entry on oeis.org

0, 1, 4, 25, 224, 2649, 38880, 679449, 13749248, 315919665, 8122432000, 231002307449, 7199799644160, 244028744225993, 8936047251296256, 351569799174274425, 14789182545666244608, 662389019735008588129, 31470659616611382460416, 1580849762199983023572313

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A010050, A279927, A293192, A302583, A302584, A302585, A302586, A302587, A302605, A302606, A302609.

Table[n! SeriesCoefficient[Exp[n x] ArcTan[x], {x, 0, n}], {n, 0, 19}]
Join[{0}, Table[n^n (HypergeometricPFQ[{1, 1, 1 - n}, {2}, -(I/n)] + HypergeometricPFQ[{1, 1, 1 - n}, {2}, I/n])/2, {n, 19}]]

a(n) ~ arctan(1) * n^n. - Vaclav Kotesovec, Jun 09 2019

a(n) = Sum_{k=1..n, k odd} (-1)^((k-1)/2)*binomial(n,k)*(k-1)!*n^(n-k). - Fabian Pereyra, Oct 05 2024

A345632 Sum of terms of even index in the binomial decomposition of n^(n-1).

Original entry on oeis.org

1, 1, 5, 28, 353, 3376, 66637, 908608, 24405761, 432891136, 14712104501, 321504185344, 13218256749601, 343360783937536, 16565151205544957, 498676704524517376, 27614800115689879553, 945381827279671853056, 59095217374989483261925, 2267322327322331161821184, 157904201452248753415276001

Offset: 1

Olivier Gérard, Jun 21 2021

When writing n^(n-1) (A000169) as a sum of powers of n using the binomial theorem, one can separately sum the even and the odd powers of n. This is the even part.

Cf. A345633 (odd part).
Cf. A062024, A302583.
Cf. A000169, A007778, A092364, A081131.

Table[Plus @@ Table[(n-1)^(2 k) Binomial[n-1, 2 k], {k, 0, Floor[n/2]}], {n, 1, 21}]

a(n+1) = Sum_{k=0..floor(n/2)} n^(2k) binomial(n, 2k).

a(n+1) = ((1 - n)^n + (1 + n)^n)/2. - Stefano Spezia, Jun 21 2021

cp's OEIS Frontend

A062024 a(n) = ((n+1)^n + (n-1)^n)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A302587 a(n) = n! * [x^n] exp(n*x)*tanh(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A302585 a(n) = n! * [x^n] exp(n*x)/cosh(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A302586 a(n) = n! * [x^n] exp(n*x)*tan(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A302605 a(n) = n! * [x^n] exp(n*x)*arcsin(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A302609 a(n) = n! * [x^n] exp(n*x)*arctanh(x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A302606 a(n) = n! * [x^n] exp(n*x)*arcsinh(x).

Views

Author

Keywords

Links

Crossrefs

A302587 a(n) = n! * [x^n] exp(nx)tanh(x).

A302586 a(n) = n! * [x^n] exp(nx)tan(x).

A302605 a(n) = n! * [x^n] exp(nx)arcsin(x).

A302609 a(n) = n! * [x^n] exp(nx)arctanh(x).

A302606 a(n) = n! * [x^n] exp(nx)arcsinh(x).

A302608 a(n) = n! * [x^n] exp(nx)arctan(x).