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Row sums of A185249.
2

%I #23 Aug 30 2023 03:51:21

%S 1,1,2,3,7,18,58,221,1055,6485,50536,478836,5440177,76190368,

%T 1363369202,31071197441,868697997215,29108097152159,1188937117381254,

%U 61492149340335133,4073478163425582443,335184988593624746984,33170032360612393972748

%N Row sums of A185249.

%C From _Vaclav Kotesovec_, Aug 27 2023: (Start)

%C a(n)^(1/n^2) tends to 1.1130228... (based on the first 750 terms).

%C Conjecture: the constant is the same as in A343032. (End)

%H Alois P. Heinz, <a href="/A186232/b186232.txt">Table of n, a(n) for n = 0..100</a>

%p with(LinearAlgebra):

%p ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:

%p T := proc(n,k)

%p if n=k then 1

%p elif irem(n+k,2)=1 then 0

%p else Determinant(Matrix((n-k)/2, (i, j)-> ctln(i+j-1+k)))

%p fi

%p end:

%p a:= n-> add(T(n,k), k=0..n):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 15 2011

%t t[n_, n_] = 1; t[n_, k_] /; Mod[n+k, 2] == 1 = 0; t[n_, k_] := Array[CatalanNumber[#1 + #2 - 1 + k]&, {(n-k)/2, (n-k)/2}] // Det; a[n_] := Sum[t[n, k], {k, 0, n}]; Table[a[n], {n, 0, 22} ] (* _Jean-François Alcover_, Mar 19 2014, after _Alois P. Heinz_ *)

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 15 2011

%E Corrected and extended by _Alois P. Heinz_, Feb 15 2011