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A322938 a(n) = binomial(2*n + 2, n + 2) - 1.

%I A322938 #23 Aug 05 2025 16:17:56
%S A322938 0,3,14,55,209,791,3002,11439,43757,167959,646645,2496143,9657699,
%T A322938 37442159,145422674,565722719,2203961429,8597496599,33578000609,
%U A322938 131282408399,513791607419,2012616400079,7890371113949,30957699535775,121548660036299,477551179875951
%N A322938 a(n) = binomial(2*n + 2, n + 2) - 1.
%H A322938 G. C. Greubel, <a href="/A322938/b322938.txt">Table of n, a(n) for n = 0..1000</a>
%F A322938 Let G(x) = (2*x^2-x+1)/(2*(x-1)*x^2)-(I*(2*x-1))/(2*x^2*sqrt(4*x-1)) with Im(x) > 0, then a(n) = [x^n] G(x). The generating function G(x) satisfies the differential equation 9*x - 16*x^2 + 4*x^3 = (8*x^5 - 22*x^4 + 21*x^3 - 8*x^2 + x)*diff(G(x), x) + (12*x^4 - 36*x^3 + 38*x^2 - 16*x + 2)*G(x).
%F A322938 From _Peter Bala_, Feb 25 2022: (Start)
%F A322938 a(n) = Sum_{k = 0..n+1} binomial(n+k,k+1).
%F A322938 a(n) = Sum_{k = 0..n-1} binomial(n+k+2,k+1).
%F A322938 More generally, Sum_{k = 0..n+m} binomial(n+k,k+1) = Sum_{k = 0..n-1} binomial(n+k+m+1,k+1) = binomial(2*n+m+1,n) - 1. (End)
%F A322938 a(n) = A001791(n+1) - 1. - _Hugo Pfoertner_, Feb 26 2022
%F A322938 a(n) = n/(n+2) * binomial(2*n+2, n+1) * Sum_{k = 0..n+1} 1/binomial(n+k+1, k). - _Peter Bala_, Aug 05 2025
%p A322938 aList := proc(len) local gf, ser; assume(Im(x) > 0);
%p A322938 gf := (2*x^2 - x + 1)/(2*(x - 1)*x^2) - (I*(2*x - 1))/(2*x^2*sqrt(4*x - 1));
%p A322938 ser := series(gf, x, len+4):
%p A322938 seq(coeff(ser, x, n), n=0..len) end: lprint(aList(25));
%t A322938 Table[Binomial[2 n + 2, n + 2] - 1, {n, 0, 25}]
%o A322938 (Magma)
%o A322938 [Binomial(2*n+2,n+2) -1: n in [0..30]]; // _G. C. Greubel_, Apr 22 2024
%o A322938 (SageMath)
%o A322938 [binomial(2*n+2,n+2) - 1 for n in range(31)] # _G. C. Greubel_, Apr 22 2024
%Y A322938 Cf. A001791, A014473, A030662 (d=0), A010763 (d=1), this sequence (d=2).
%K A322938 nonn,easy
%O A322938 0,2
%A A322938 _Peter Luschny_, Feb 13 2019