A003149 -id:A003149 - cp's OEIS frontend

A353250 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); numerators.

1, 1, 4, 24, 48, 480, 960, 13440, 26880, 161280, 322560, 7096320, 14192640, 369008640, 738017280, 295206912, 590413824, 20074070016, 40148140032, 1525629321216, 15256293212160, 30512586424320, 61025172848640, 2807157951037440, 5614315902074880

Offset: 0

Vladimir Reshetnikov, Apr 08 2022

			a(0) = 1,
a(1) = 2/(1/1 + 1/1) = 1,
a(2) = 2/(1/1 + 1/2) = 4/3,
a(3) = 2/(1/(4/3) + 1/3) = 24/13,
a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
This sequence gives the numerators: 1, 1, 4, 24, 48, ...

Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Lerch Transcendent.
Wikipedia, Harmonic mean.

Cf. A353251 (denominators).

Cf. A003149, A136128, A191778 (has many terms in common), A241519, A242376.

Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 24}] // Numerator (* or *)
a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 24}] // Numerator

a(n) = numerator(1/(1/2^n - Re(Phi(2, 1, n+1)))), where Phi(z, s, a) is the Lerch transcendent.

A353251 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); denominators.

Original entry on oeis.org

1, 1, 3, 13, 19, 143, 223, 2521, 4201, 21563, 37691, 737161, 1328521, 31463413, 57821173, 21404465, 39854897, 1267947073, 2383173185, 85428430547, 808549483039, 1535039635999, 2921975382559, 128230606647497, 245195521274057, 2348840786785261, 4508193056814061

Offset: 0

Vladimir Reshetnikov, Apr 08 2022

			a(0) = 1,
a(1) = 2/(1/1 + 1/1) = 1,
a(2) = 2/(1/1 + 1/2) = 4/3,
a(3) = 2/(1/(4/3) + 1/3) = 24/13,
a(4) = 2/(1/(24/13) + 1/4) = 48/19, etc.
This sequence gives the denominators: 1, 1, 3, 13, 19, ...

Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Lerch Transcendent.
Wikipedia, Harmonic mean.

Cf. A353250 (numerators).

Cf. A003149, A136128, A241519, A242376.

Table[1/(1/2^n - Re[LerchPhi[2, 1, n + 1]]), {n, 0, 26}] // Denominator (* or *)
a[0] = 1; a[n_Integer] := a[n] = 2/(1/a[n-1] + 1/n); Table[a[n], {n, 0, 26}] // Denominator

a(n) = denominator(1/(1/2^n - Re(Phi(2, 1, n+1)))), where Phi(z, s, a) is the Lerch transcendent.

A358791 a(n) = n!Sum_{m=0..floor(n/2)} binomial(n,2m)^(-1).

Original entry on oeis.org

1, 1, 4, 8, 52, 156, 1536, 6144, 84096, 420480, 7453440, 44720640, 974972160, 6824805120, 176504832000, 1412038656000, 42224136192000, 380017225728000, 12893605517721600, 128936055177216000, 4892595136708608000

Offset: 0

Vladimir Kruchinin, Dec 01 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A003149.

a(n):=n!*sum(1/binomial(n,2*m),m,0,floor(n/2));

a(n) = n!*sum(m=0, n\2, 1/binomial(n, 2*m)); \\ Michel Marcus, Dec 01 2022

E.g.f.: (1/2)*( log(1+x)/x^2-log(1-x)*(x^2+4*x-4)/(x^4-4*x^3+4*x^2)+6/(x^3-2*x^2-x+2) ).

P-recursive: 2*n*(n + 2)*(n - 2)*(3*n - 2)*a(n) = n*(n + 1)*(n + 2)*(n - 2)*(3*n - 2)*a(n-1) + 2*n^3*(n - 1)*(n - 2)*(3*n + 4)*a(n-2) - n^3*(n - 1)^2*(n - 2)*(3*n + 4)*a(n-3) with a(0) = a(1) = 1 and a(2) = 4. - Peter Bala, Apr 13 2023

A366565 Decimal expansion of the smaller real solution to x*2^(1/x) = e.

Original entry on oeis.org

3, 2, 7, 5, 6, 2, 4, 1, 3, 9, 7, 7, 5, 1, 6, 9, 4, 0, 0, 9, 2, 8, 2, 0, 8, 1, 2, 5, 9, 9, 1, 2, 2, 0, 4, 4, 3, 3, 9, 6, 4, 4, 6, 9, 6, 6, 5, 4, 2, 2, 7, 4, 2, 0, 4, 2, 9, 6, 9, 6, 9, 5, 4, 9, 6, 3, 4, 7, 6, 6, 3, 1, 4, 2, 2, 3, 3, 8, 7, 4, 9, 7, 5, 4, 6, 7, 9, 4, 2

Offset: 0

Hugo Pfoertner, Oct 23 2023

This is the constant alpha occurring in the asymptotic analysis of random walks on the hypercube (Lemma 3, page 7, attributed to Bjorn Poonen), in Diaconis, Graham, and Morrison (1988). See link for more information.

			0.32756241397751694009282081259912204433964469665422742...

Persi Diaconis, R. L. Graham, and J. A. Morrison, Asymptotic Analysis of a Random Walk on a Hypercube with Many Dimensions, Technical Report EFS NFS 307, Department of Statistics, Stanford University, December 1988.
Persi Diaconis, R. L. Graham, and J. A. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions, Random Structures & Algorithms, Volume 1, Issue 1, Pages 51-72, Spring 1990.
Gordon Slade, Self-avoiding walk on the hypercube, Random Structures & Algorithms, Volume 62, Issue 3, May 2023, Pages 689-736.

Cf. A001113, A003149, A324495.

RealDigits[-Log[2]/ProductLog[-1, -Log[2]/E], 10, 120][[1]] (* Vaclav Kotesovec, Nov 03 2023 *)

PARI
```
solve (x = 0.3, 0.35, x*2^(1/x)-exp(1))
```

Equals -log(2)/LambertW(-1, -log(2)/exp(1)). - Vaclav Kotesovec, Nov 03 2023

A299121 a(n) = Sum_{k=0..n} (k*(n-k))!.

Original entry on oeis.org

1, 2, 3, 6, 38, 1490, 443762, 965262242, 23539096637282, 4878608938121399042, 16752209028723653862101762, 531115497554502361264846265433602, 392660148984369152453298787770243889113602, 2811644066816242246665284589590844386691155533363202

Offset: 0

Vaclav Kotesovec, Feb 03 2018

nonn

Cf. A000142, A003149, A003422.

Table[Sum[(k*(n-k))!, {k, 0, n}], {n, 0, 14}]

a(n) ~ sqrt(Pi) * n^(n^2/2 + 1) / (2^(n^2/2 + 1/2) * exp(n^2/4)) if n is even, and a(n) ~ sqrt(Pi) * n^(n^2/2 + 1/2) / (2^(n^2/2 - 1) * exp(n^2/4)) if n is odd.

A368209 a(n) = Sum_{k=0..n} BarnesG(k)*BarnesG(n-k).

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 29, 604, 69724, 49836144, 250872492816, 10113420362487552, 3669877057922582621184, 13317216838086531218401935360, 531580547910000731718546175028428800, 254627927130379381409123944181515703549952000

Offset: 0

Vaclav Kotesovec, Dec 17 2023

nonn

Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A000178, A003149.

Table[Sum[BarnesG[k]*BarnesG[n-k], {k, 0, n}], {n, 0, 15}]

a(n) ~ 2^(n/2) * Pi^(n/2 - 1) * n^(n^2/2 - 2*n + 23/12) / (A * exp(3*n^2/4 - 2*n - 1/12)), where A = A074962 is the Glaisher-Kinkelin constant.

cp's OEIS Frontend

A353250 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); numerators.

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A353251 a(0) = 1, a(n) = harmonic_mean(a(n-1), n), where harmonic_mean(p, q) = 2/(1/p + 1/q); denominators.

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A358791 a(n) = n!*Sum_{m=0..floor(n/2)} binomial(n,2*m)^(-1).

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Maxima

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A366565 Decimal expansion of the smaller real solution to x*2^(1/x) = e.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A299121 a(n) = Sum_{k=0..n} (k*(n-k))!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A368209 a(n) = Sum_{k=0..n} BarnesG(k)*BarnesG(n-k).

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Author

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Crossrefs

Programs

Mathematica

Formula

A358791 a(n) = n!Sum_{m=0..floor(n/2)} binomial(n,2m)^(-1).