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 A025591 Maximal coefficient of Product_{k<=n} (1 + x^k). Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1. 34
 1, 1, 1, 2, 2, 3, 5, 8, 14, 23, 40, 70, 124, 221, 397, 722, 1314, 2410, 4441, 8220, 15272, 28460, 53222, 99820, 187692, 353743, 668273, 1265204, 2399784, 4559828, 8679280, 16547220, 31592878, 60400688, 115633260, 221653776, 425363952, 817175698 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS If k is allowed to approach infinity, this gives the partition numbers A000009. a(n) is the maximal number of subsets of {1,2,...,n} that share the same sum. LINKS T. D. Noe, Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz) Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.4. Vlad-Florin Dragoi, Valeriu Beiu, Fast Reliability Ranking of Matchstick Minimal Networks, arXiv:1911.01153 [cs.DM], 2019. Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author] E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), Article 00.2.5. Marco Mondelli, SH Hassani, R Urbanke, Construction of Polar Codes with Sublinear Complexity, arXiv preprint arXiv:1612.05295 [cs.IT], 2016-2017. See Sect. I. Robert A. Proctor, Solution of two difficult combinatorial problems with linear algebra, American Mathematical Monthly 89, 721-734. B. D. Sullivan, On a conjecture of Adrica and Tomescu, J. Int. Sequences 16 (2013), Article 13.3.1. FORMULA a(n) = A063865(n) + A063866(n). a(n) ~ sqrt(6/Pi) * 2^n / n^(3/2) [conjectured by Andrica and Tomescu (2002) and proved by Sullivan (2013)]. - Vaclav Kotesovec, Mar 17 2020 MAPLE b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,       `if`(i=0, 1, b(n+i, i-1)+b(abs(n-i), i-1)))     end: a:=n-> b(0, n)+b(1, n): seq(a(n), n=0..40);  # Alois P. Heinz, Mar 10 2014 MATHEMATICA f[n_, s_] := f[n, s]=Which[n==0, If[s==0, 1, 0], Abs[s]>(n*(n+1))/2, 0, True, f[n-1, s-n]+f[n-1, s+n]]; Table[Which[Mod[n, 4]==0||Mod[n, 4]==3, f[n, 0], Mod[n, 4]==1||Mod[n, 4]==2, f[n, 1]], {n, 0, 40}] {* Second program: *) p = 1; Flatten[{1, Table[p = Expand[p*(1 + x^n)]; Max[CoefficientList[p, x]], {n, 1, 50}]}] (* Vaclav Kotesovec, May 04 2018 *) b[n_, i_] := b[n, i] = If[n > i(i+1)/2, 0, If[i == 0, 1, b[n+i, i-1] + b[Abs[n-i], i-1]]]; a[n_] := b[0, n] + b[1, n]; a /@ Range[0, 40] (* Jean-François Alcover, Feb 17 2020, after Alois P. Heinz *) PROG (PARI) a(n)=if(n<0, 0, polcoeff(prod(k=1, n, 1+x^k), n*(n+1)\4)) CROSSREFS Cf. A039828, A063865, A069918, A063866, A063867, A083309, A083527, A086376. Sequence in context: A236393 A350504 A039822 * A028409 A348850 A183559 Adjacent sequences:  A025588 A025589 A025590 * A025592 A025593 A025594 KEYWORD nonn,nice AUTHOR STATUS approved

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Last modified May 28 16:37 EDT 2022. Contains 354119 sequences. (Running on oeis4.)