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A350653 a(n) is the number of weak compositions of n into n-1 parts in which at least one part is zero.

%I A350653 #21 Mar 06 2022 09:02:34
%S A350653 0,2,12,52,205,786,2996,11432,43749,167950,646635,2496132,9657687,
%T A350653 37442146,145422660,565722704,2203961413,8597496582,33578000591,
%U A350653 131282408380,513791607399,2012616400058,7890371113927,30957699535752,121548660036275
%N A350653 a(n) is the number of weak compositions of n into n-1 parts in which at least one part is zero.
%F A350653 a(n) = binomial(2*n-2,n) - (n-1) = A001791(n-1) -n+1.
%F A350653 G.f.: 4*x^2/((1 - sqrt(1 - 4*x))^2*sqrt(1 - 4*x)) - (1 - 2*x + 2*x^2)/(1 - x)^2. - _Stefano Spezia_, Jan 10 2022
%F A350653 D-finite with recurrence +n*(11*n-38)*a(n) -(n-1)*(73*n-244)*a(n-1) +2*(67*n^2-364*n+492)*a(n-2) -4*(9*n-22)*(2*n-7)*a(n-3)=0. - _R. J. Mathar_, Mar 06 2022
%e A350653 a(5)=52 since 5 can be written as 5+0+0+0 (4 such compositions); 4+1+0+0 (12 such compositions); 3+2+0+0 (12 such compositions); 3+1+1+0 (12 such compositions); 2+2+1+0 (12 such compositions). All these weak compositions contain at least one zero.
%t A350653 a[n_] := Binomial[2*n - 2, n] - n + 1; Array[a, 25, 2] (* _Amiram Eldar_, Jan 10 2022 *)
%Y A350653 Cf. A001791, A010763.
%K A350653 nonn,easy
%O A350653 2,2
%A A350653 _Enrique Navarrete_, Jan 09 2022