Shalosh B. Ekhad - cp's OEIS frontend

User: Shalosh B. Ekhad

Shalosh B. Ekhad's wiki page.

Shalosh B. Ekhad has authored 2 sequences.

A182523 Rademacher's sequence C_{011}(N) times (2n)!, where C_{011}(N) is the coefficient of 1/(q-1) in the partial fraction decomposition of 1/((1-q)(1-q^2)...(1-q^N)).

Original entry on oeis.org

-2, -6, -170, -9520, -874902, -118950678, -22370367448, -5550123527520

Offset: 1

Shalosh B. Ekhad, May 03 2012

Hans Rademacher conjectured that C_{011}(N) converge to -0.292927573960. This conjecture is false.

Named after the German-American mathematician Hans Adolph Rademacher (1892-1969). - Amiram Eldar, Jun 22 2021

			For n=1, the coefficient of 1/(q-1) in the partial fraction decomposition of 1/(1-q) is -1, multiplied by 2! this gives -2.

Hans Rademacher, Topics in Analytic Number Theory, Springer, 1973, p. 302.

Andrew V. Sills and Doron Zeilberger, Rademacher's infinite partial fraction conjecture is (almost certainly) false, arXiv:1110.4932v1 [math.NT], 2011.
Andrew V. Sills and Doron Zeilberger, Rademacher's Infinite Partial Fraction Conjecture is (almost certainly) False, Oct 21 2011; Local copy, pdf file only, no active links.
Andrew V. Sills and Doron Zeilberger, HANS (maple package); Local copy.

Maple
```
See above link to HANS (maple package).
```

See above article for an efficient recurrence.

A028420 Number of monomer-dimer tilings of n X n chessboard.

Original entry on oeis.org

1, 1, 7, 131, 10012, 2810694, 2989126727, 11945257052321, 179788343101980135, 10185111919160666118608, 2172138783673094193937750015, 1743829823240164494694386437970640, 5270137993816086266962874395450234534887, 59956919824257750508655631107474672284499736089

Offset: 0

Jennifer Henry, Shalosh B. Ekhad, and Steven Finch

Also the total number of matchings (not necessarily perfect ones; i.e., Hosoya index) in the n X n grid. - Andre Poenitz (poenitz(AT)htwm.de), Nov 20 2003

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 406-412.

Jennifer Henry, Table of n, a(n) for n = 0..21 [From S. R. Finch, Jan 30 2009]
J. H. Ahrens, Paving the chessboard. J. Combin. Theory Ser. A 31(1981), no. 3, 277--288. MR0635371 (84d:05009). See Table I. - _N. J. A. Sloane_, Mar 27 2012
Steven R. Finch, Two Dimensional Monomer-Dimer Constant [Broken link]
Steven R. Finch, Two Dimensional Monomer-Dimer Constant [From the Wayback machine] H P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 362.
Svenja Huntemann and Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020).
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hosoya Index
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
D. Zeilberger, Source; Local copy
Index entries for sequences related to dominoes

Cf. A004003. A diagonal of A210662.

Row sums of A242861.

b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l)>0 then (t-> b(n-t, map(h->h-t, l)))(min(l))
    else for k while l[k]>0 do od; `if`(k b(n, [0$n]):
seq(a(n), n=0..13);  # Alois P. Heinz, Dec 04 2020

Table[With[{g = GridGraph[{n, n}]}, Count[Subsets[EdgeList[g], Length @ Flatten @ FindIndependentEdgeSet[g]], ?(IndependentEdgeSetQ[g, #] &)]], {n, 4}] (* _Eric W. Weisstein, May 28 2017 *)
b[n_, l_] := b[n, l] = Module[{k}, Which[
     n == 0, 1,
     Min[l] > 0, Function[t, b[n-t, Map[#-t&, l]]][Min[l]],
     True, For[k = 1, l[[k]] > 0, k++]; If[k < Length[l] &&
          l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0] +
          Sum[If[n j]]], {j, 1, 2}]]];
a[n_] := b[n, Table[0, {n}]];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 30 2021, after Alois P. Heinz *)

Broken links corrected by Steven Finch, Jan 27 2009

a(0)=1 prepended by Alois P. Heinz, Dec 04 2020

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User: Shalosh B. Ekhad

A182523 Rademacher's sequence C_{011}(N) times (2n)!, where C_{011}(N) is the coefficient of 1/(q-1) in the partial fraction decomposition of 1/((1-q)(1-q^2)...(1-q^N)).

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