A002897 -id:A002897 - cp's OEIS frontend

A359532 Decimal expansion of 2*log(2)/Pi.

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

Offset: 0

Stefano Spezia, Jan 04 2023

2*log(2)*n/Pi is also the dominant term in the asymptotic expansion of Sum_{k=1..n-1} (-1)^(k+1)*csc(Pi*k/n) at n tending to infinity. - Iaroslav V. Blagouchine, Apr 10 2025

			0.441271200305303186792912864235995381965...

Iaroslav V. Blagouchine, On a Generalization of Watson's Trigonometric Sum (On Dowker's Sum of Order One Half), INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 25, Article #A30, pp. 1-33, 2025. See p. 18.
John M. Campbell, Applications of a class of transformations of complex sequences, arXiv:2212.13305 [math.NT], 2022. See pp. 2, 8.

Cf. A000796, A001008, A002162, A002805, A002897, A016627, A016813, A060294.

Cf. A359533.

Mathematica
```
First[RealDigits[N[2Log[2]/Pi,98]]]
```

Equals A016627/A000796.
Equals A002162*A060294.
Equals 2*A284983.
Equals Sum_{i>=0} (-1/64)^i*binomial(2*i, i)^3*(4*i + 1)*H_{2*i}, where H_m is the m-th harmonic number (negated).

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 04 2023

			0.276427204245986573092639825616889946783740795...

John M. Campbell, Applications of a class of transformations of complex sequences, arXiv:2212.13305 [math.NT], 2022. See p. 6.

Cf. A000796, A001008, A002805, A002897, A016639, A016813, A063448, A068465, A068466, A091925, A203142, A203143.

Cf. A359532.

First[RealDigits[N[(Gamma[1/8]Gamma[3/8]/(Gamma[1/4]Gamma[3/4]))^2/(6Sqrt[2]Pi)-4Log[2]/Pi,100]]]

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8)/(Gamma(1/4)*Gamma(3/4)))^2/(6*sqrt(2)*Pi).

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8))^2/(12*sqrt(2)*Pi^3).

A374605 a(n) = Sum_{k = 0..n} binomial(n, k)^2binomial(n+k, k)binomial(3n+2k, n).

Original entry on oeis.org

1, 13, 621, 40864, 3116125, 258687513, 22695228864, 2069939892096, 194303918495709, 18648446389798225, 1821631879087498621, 180513102382789033728, 18101940249015916366528, 1833572727177462316881472, 187323995560940882748187200, 19279943156312884441303524864, 1997221716775275248175573251037

Offset: 0

Peter Bala, Jul 20 2024

Compare with the identity Sum_{k = 0..n} binomial(n, k)^2 * binomial(n+k, k) * binomial(2*n+k, n) = binomial(2*n, n)^3 = A002897(n).
It is easy to see that for odd prime p, binomial(2*n, n)^3 is divisible by p^3 for integer n in the interval [(p + 1)/2, p - 1]. A similar property appears to hold for the present sequence. We conjecture that for prime p >= 5, a(n) is divisible by p^3 for integer n in the interval [ceiling((2*p + 1)/3), p - 1] (checked up to p = 101).
More generally, for m >= 2, a similar divisibility property appears to hold for the sequence whose n-th term is equal to Sum_{k = 0..n} binomial(n, k)^2* binomial(n+k, k)*binomial((m + 1)*n + m*k, n).

			Factorization of a(8) thru a(10) showing divisibility by 11^3:
a(8) = (3^6)*11^3*10667*18773
a(9) = (5^2)*7*(11^3)*(13^3)*3607*10103
a(10) = (11^3)*(13^4)*31*22699*68099.

Cf. A176285, A374606.

seq(add(binomial(n, k)^2*binomial(n+k, k)*binomial(3*n+2*k, n), k = 0..n), n = 0..20);
# faster program for large n
seq(simplify(binomial(3*n, n)*hypergeom([-n, -n, (3*n+1)/2, (3*n+2)/2], [1, 1, n+1/2], 1)), n = 0..20);

a(n) = binomial(3*n, n)*hypergeom([-n, -n, (3*n+1)/2, (3*n+2)/2], [1, 1, n+1/2], 1).
P-recursive: 16*n^3*(5616*n^4 - 30888*n^3 + 63459*n^2 - 57709*n + 19600)*(4*n - 1)^2*(4*n - 3)^2*a(n) = 36*(72783360*n^11 - 655050240*n^10 + 2595613248*n^9 - 5966404272*n^8 + 8824615470*n^7 - 8803399545*n^6 + 6034085115*n^5 - 2836309905*n^4 + 893904075*n^3 - 179376410*n^2 + 20562360*n - 1019200)*a(n-1) +  27*n*(5616*n^4 - 8424*n^3 + 4491*n^2 - 991*n + 78)*(3*n - 4)^3*(3*n - 5)^3*a(n-2) with a(0) = 1, a(1) = 13.
a(n) ~ 3^(9*n/2) * (1 + sqrt(3))^(6*n + 3) / (Pi^(3/2) * n^(3/2) * 2^(9*n + 9/2)). - Vaclav Kotesovec, Jul 22 2024

A229955 Triangular array read by rows: 3 dimensional analog of A227997.

Original entry on oeis.org

8, 152, 64, 5056, 2432, 512, 205720, 104000, 29184, 4096, 9305152, 4828544, 1525248, 311296, 32768, 449404224, 236984448, 79898624, 19226624, 3112960, 262144, 22695553536, 12099474432, 4251479040, 1123909632, 221839360, 29884416, 2097152, 1183891745688, 636162156096, 230017430016, 64636047360, 14330265600, 2413559808, 278921216, 16777216

Offset: 1

Geoffrey Critzer, Oct 04 2013

T(n,k) is the number of walks on the 3 dimensional grid that start and end at the origin using 2n steps and having exactly k primitive loops. The steps are in the eight directions: (1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1), (-1,1,1), (-1,1,-1), (-1,-1,1), (-1,-1,-1).  A primitive loop is a walk that starts and ends on the origin but does not otherwise touch the origin.
Column 1 is A094059.
Row sums are A002897.

			8,
152, 64,
5056, 2432, 512,
205720, 104000, 29184, 4096,
9305152, 4828544, 1525248, 311296, 32768,
449404224, 236984448, 79898624, 19226624, 3112960, 262144

nn=6;a=Sum[Binomial[2n,n]^3x^n,{n,0,nn}];Map[Select[#,#>0&]&,Drop[CoefficientList[Series[1/(1-y(1-1/a)),{x,0,nn}],{x,y}],1]]//Grid

G.f.: 1/( 1 - y*(1 - 1/A(x)) ) where A(x) is the o.g.f. for A002897.

Generally for such walks in N dimensions: 1/( 1 - y*(1 - 1/B(x)) ) where B(x) = Sum_{n>=0} binomial(2n,n)^N*x^n.

A302577 Convolution square root of A186284 multiplied by 2^n.

Original entry on oeis.org

1, 2, 94, 6628, 554246, 50936956, 4971074892, 505747739784, 53048521913478, 5695802803696236, 622942370315360004, 69155891028297395448, 7772714892571857579036, 882718626126348791323992, 101137353917153181195426264, 11676481964194514316750017040

Offset: 0

Vaclav Kotesovec, Apr 10 2018

nonn

Cf. A186284, A127776, A002897.

nmax = 20; CoefficientList[Series[Hypergeometric2F1[1/4, 1/4, 1, 64*x]^(1/4), {x, 0, nmax}], x] * 2^Range[0, nmax]

a(n) ~ Pi^(9/8) * 2^(7*n - 5/4) / (Gamma(1/4)^(7/2) * n^(3/2)).

A376228 a(n) = (6n+1) (2*n)!^3 / n!^6.

Original entry on oeis.org

1, 56, 2808, 152000, 8575000, 496093248, 29188893888, 1738242215424, 104455598247000, 6321316756040000, 384702925005146176, 23520160000755565056, 1443504313932496274368, 88879637239345064000000, 5487711609457595160000000, 339644002672064899081728000, 21065385579274083203741943000

Offset: 0

Paul D. Hanna, Oct 17 2024

nonn

			G.f.: A(x) = 1 + 56*x + 2808*x^2 + 152000*x^3 + 8575000*x^4 + 496093248*x^5 + 29188893888*x^6 + 1738242215424*x^7 + ...
where
A(x) = 1 + 7*(1/2)^3*64*x + 13*((1*3)/(2*4))^3*64^2*x^2 + 19*((1*3*5)/(2*4*6))^3*64^3*x^3 + 25*((1*3*5*7)/(2*4*6*8))^3*64^4*x^4 + ... + (6*n+1)*(2*n)!^3/n!^6*x^n + ...
SPECIFIC VALUES.
At x = 1/256 we have the series
4/Pi = 1 + 7*(1/2)^3/4 + 13*((1*3)/(2*4))^3/4^2 + 19*((1*3*5)/(2*4*6))^3/4^3 + 25*((1*3*5*7)/(2*4*6*8))^3/4^4 + ... = 1.273239544735162686...
see formula 28 in the Ramanujan link for details.

S. Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, p. 45.

Cf. A002897.

a[n_]:=(6*n+1) * (2*n)!^3 / n!^6; Array[a,17,0] (* Stefano Spezia, Oct 17 2024 *)

a(n) = (6*n+1) * A002897(n).
a(n) ~ 3*2^(6*n+1)/sqrt(n*Pi^3). - Stefano Spezia, Oct 17 2024
D-finite with recurrence n^3*a(n) +8*(56*n^3-252*n^2+330*n-141)*a(n-1) -4096*(2*n-3)^3*a(n-2)=0. - R. J. Mathar, Oct 24 2024

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A359532 Decimal expansion of 2*log(2)/Pi.

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A359533 Decimal expansion of Sum_{k>=0} (-1/64)^k*binomial(2*k, k)^3*(4*k + 1)*H_k, where H_k is the k-th harmonic number (negated).

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A374605 a(n) = Sum_{k = 0..n} binomial(n, k)^2*binomial(n+k, k)*binomial(3*n+2*k, n).

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A229955 Triangular array read by rows: 3 dimensional analog of A227997.

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A302577 Convolution square root of A186284 multiplied by 2^n.

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Mathematica

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A376228 a(n) = (6*n+1) * (2*n)!^3 / n!^6.

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A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).

A374605 a(n) = Sum_{k = 0..n} binomial(n, k)^2binomial(n+k, k)binomial(3n+2k, n).

A376228 a(n) = (6n+1) (2*n)!^3 / n!^6.