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 A242105 Number of sequences (x(k))_{k=1..n}, of n strictly increasing terms of nonnegative integers {x(1)
 1, 2, 7, 44, 428, 5802, 102322, 2239844, 58849332, 1810039960, 63930543419, 2553881719348, 113979459829296, 5625823639358928, 304505544257483550, 17944306197698666740, 1144180970802458374244, 78517953136289477587608, 5771772521253777092098050 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also the first occurring nonzero terms in rows of triangle A261897, without repetitions. - Reinhard Zumkeller, Sep 06 2015 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 L. Haddad and C. Helou, Finite Sequences Dominated by the Squares, Journal of Integer Sequences, Volume 18, 2015, Issue 1, Article 15.1.8. FORMULA a(n) = Sum_{k=1..n} (-1)^{k-1}* C((n-k+1)^2+k-1,k) * a(n-k), for n>1. a(n) = C(n^2,n-1) + C(n^2-1,n-1) - Sum_{k=2..n-1} C(n^2-k^2,n-k+1) *a(k-1), for n>1. Conjecture: lim n->infinity a(n)^(1/n)/n = 2. - Vaclav Kotesovec, Feb 26 2017 EXAMPLE For n=2 the a(2) = 7 solutions are (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4). MAPLE a:= proc(n) option remember; `if`(n<2, n+1, add((-1)^(k-1)* binomial((n-k+1)^2+k-1, k) * a(n-k), k=1..n)) end: seq(a(n), n=0..25); # Alois P. Heinz, Aug 15 2014 MATHEMATICA a[0] = 1; a[1] = 2; a[n_] := a[n] = Sum[(-1)^(k-1)*Binomial[(n-k+1)^2+k-1, k]*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 26 2017 *) CROSSREFS Cf. A261897. Sequence in context: A128579 A194453 A346258 * A001046 A158257 A348857 Adjacent sequences: A242102 A242103 A242104 * A242106 A242107 A242108 KEYWORD nonn AUTHOR Charles Helou, Aug 14 2014 STATUS approved

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