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A242105 Number of sequences (x(k))_{k=1..n}, of n strictly increasing terms of nonnegative integers {x(1) satisfying x(k) <= k^2 for all k.

%I A242105 #33 Jul 22 2020 11:42:35
%S A242105 1,2,7,44,428,5802,102322,2239844,58849332,1810039960,63930543419,
%T A242105 2553881719348,113979459829296,5625823639358928,304505544257483550,
%U A242105 17944306197698666740,1144180970802458374244,78517953136289477587608,5771772521253777092098050
%N A242105 Number of sequences (x(k))_{k=1..n}, of n strictly increasing terms of nonnegative integers {x(1)<x(2)<...<x(n)}, satisfying x(k) <= k^2 for all k.
%C A242105 Also the first occurring nonzero terms in rows of triangle A261897, without repetitions. - _Reinhard Zumkeller_, Sep 06 2015
%H A242105 Alois P. Heinz, <a href="/A242105/b242105.txt">Table of n, a(n) for n = 0..200</a>
%H A242105 L. Haddad and C. Helou, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Helou/helou7.html">Finite Sequences Dominated by the Squares</a>, Journal of Integer Sequences, Volume 18, 2015, Issue 1, Article 15.1.8.
%F A242105 a(n) = Sum_{k=1..n} (-1)^{k-1}* C((n-k+1)^2+k-1,k) * a(n-k), for n>1.
%F A242105 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.
%F A242105 Conjecture: lim n->infinity a(n)^(1/n)/n = 2. - _Vaclav Kotesovec_, Feb 26 2017
%e A242105 For n=2 the a(2) = 7 solutions are (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4).
%p A242105 a:= proc(n) option remember; `if`(n<2, n+1, add((-1)^(k-1)*
%p A242105        binomial((n-k+1)^2+k-1, k) * a(n-k), k=1..n))
%p A242105     end:
%p A242105 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 15 2014
%t A242105 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 *)
%Y A242105 Cf. A261897.
%K A242105 nonn
%O A242105 0,2
%A A242105 _Charles Helou_, Aug 14 2014