Tengiz Gogoberidze - cp's OEIS frontend

User: Tengiz Gogoberidze

Tengiz Gogoberidze has authored 2 sequences.

A367110 Decimal expansion of Sum_{k has exactly 3 bits equal to 1 in base 2} 1/k.

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

Offset: 1

Tengiz Gogoberidze, Dec 16 2023

For 1 bit equal to 1 the sum is 2, for 2 bits equal to 1 the sum is 1.52899956069688841838263949451... (see A179951).

			1.4285915458526381...

Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Cf. A179951, A014311.

RealDigits[iSum[1, 3, 105, 2]][[1]] (* Amiram Eldar, Dec 16 2023, using Baillie's irwinSums.m *)

Equals Sum_{m>=2} Sum_{j=1..m-1} Sum_{i=0..j-1} 1/(2^i + 2^j + 2^m).
Equals 2 * Sum_{j>=2} Sum_{i=1..j-1} 1/(2^i + 2^j + 1).
Equals Sum_{k>=1} 1/A014311(k).

A282010 Number of ways to partition Turan graph T(2n,n) into connected components.

Original entry on oeis.org

1, 1, 12, 163, 3411, 97164, 3576001, 163701521, 9064712524, 594288068019, 45352945127123, 3973596101084108, 395147058261233761, 44170986458602383553, 5504694207040057913164, 759355292729159336345955, 115228949414563130433140659, 19129024114529146183236435660

Offset: 0

Tengiz Gogoberidze, Feb 04 2017

Turan graph T(2n,n) is also called cocktail party graph, so a(n) is the number of ways to seat n married couples for one or a few tables in such a manner that no table is fully occupied by any couple.

If we dissect (n-1)-skeleton of n-cube along some (n-2)-edges into some parts, then a(n) is the number of ways of such dissections.

			For n=1, Turan graph T(2,1) (2-empty graph) shall be partitioned into two singleton subgraphs (1 way), a(1)=1.
For n=2, Turan graph T(4,2) (square graph) shall be partitioned into: the same square graph (1 way) or one singleton + one 3-path subgraphs (4 ways) or two singleton + one 2-path subgraphs (4 ways) or two 2-path subgraphs (2 ways) or four singleton subgraphs (1 way), a(2)=12.

Indranil Ghosh, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Turan Graph

Cf. A000110, A020557

A282010 := proc(n)
    add((-1)^(n-j)*combinat[bell](2*j)*binomial(n,j),j=0..n) ;
end proc:
seq(A282010(n),n=0..20) ; # R. J. Mathar, Jun 27 2024

a[n_]:=BellB[2n];Table[Sum[((-1)^(n-j))*a[j]*Binomial[n,j],{j,0,n}],{n,0,17}] (* Indranil Ghosh, Feb 25 2017 *)

bell(n) = polcoeff( sum( k=0, n, prod(i=1, k, x/(1 - i*x)), x^n * O(x)), n)
a(n) = sum(j=0, n, ((-1)^(n-j))*bell(2*j)*binomial(n,j)); \\ Michel Marcus, Feb 05 2017

a(n) = Sum_{j=0..n} ((-1)^(n-j))*A020557(j)*binomial(n,j).

a(n) = Sum_{j=0..n} ((-1)^(n-j))*A000110(2*j)*binomial(n,j).

More terms from Michel Marcus, Feb 05 2017

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User: Tengiz Gogoberidze

A367110 Decimal expansion of Sum_{k has exactly 3 bits equal to 1 in base 2} 1/k.

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