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A325983 Row sums of the triangle A325982.

%I A325983 #31 Sep 08 2022 08:46:24
%S A325983 1,1,2,2,5,5,18,21,77,102,337,480,1449,2155,6107,9348,25355,39639,
%T A325983 104188,165596,425156,684926,1726737,2813582,6990175,11501905,
%U A325983 28232753,46854161,113841632,190362483,458480128,771855377,1844765161,3124639626,7417428613,12633074088
%N A325983 Row sums of the triangle A325982.
%H A325983 Stefano Spezia, <a href="/A325983/b325983.txt">Table of n, a(n) for n = 1..2000</a>
%F A325983 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n - 1, k - 1) - binomial(n - k - 1, k - 1) + 1.
%F A325983 a(n) = Sum_{k=0..A004526(n-1)} A007318(n - 1, k - 1) - A007318(n - k - 1, k - 1) + 1.
%F A325983 Conjecture: a(n) ~ 2^(n-2). - _Stefano Spezia_, Jun 16 2021
%F A325983 This conjecture is true. Its proof follows from _Vaclav Kotesovec_'s formula used in the 2nd Mathematica code. - _Stefano Spezia_, Jun 27 2021
%F A325983 Recurrence: n*(15*n^6 - 600*n^5 + 9874*n^4 - 85078*n^3 + 403791*n^2 - 1000762*n + 1013720)*a(n) = (45*n^7 - 1815*n^6 + 30507*n^5 - 273677*n^4 + 1393992*n^3 - 3930412*n^2 + 5362320*n - 2240000)*a(n-1) + 2*(30*n^7 - 1230*n^6 + 20813*n^5 - 186314*n^4 + 944533*n^3 - 2692476*n^2 + 3985604*n - 2395680)*a(n-2) - (255*n^7 - 10470*n^6 + 180608*n^5 - 1685204*n^4 + 9136801*n^3 - 28646510*n^2 + 47833000*n - 32569600)*a(n-3) + (15*n^7 - 585*n^6 + 11089*n^5 - 129179*n^4 + 931040*n^3 - 3910812*n^2 + 8540192*n - 7271040)*a(n-4) + 2*(165*n^7 - 6810*n^6 + 118529*n^5 - 1121692*n^4 + 6211866*n^3 - 20094734*n^2 + 35150196*n - 25693920)*a(n-5) - 4*(15*n^7 - 600*n^6 + 10309*n^5 - 98168*n^4 + 555848*n^3 - 1857428*n^2 + 3364504*n - 2524480)*a(n-6) - 8*(n-7)*(15*n^6 - 510*n^5 + 7099*n^4 - 51282*n^3 + 202026*n^2 - 411828*n + 340960)*a(n-7). - _Vaclav Kotesovec_, Jun 20 2021
%p A325983 a := n -> add(binomial(n-1, k-1)-binomial(n-k-1, k-1)+1, k = 0 .. floor((n-1)/2)): seq(a(n), n = 1 .. 40);
%t A325983 T[n_,k_]:=Binomial[n-1,k-1]-Binomial[n-k-1,k-1]+1; a[n_]:=Sum[T[n,k],{k,0,Floor[(n-1)/2]}]; Array[a,40]
%t A325983 Table[If[EvenQ[n], 2^(n - 2) + n/2 + 1 - Binomial[n, n/2]/2 + Fibonacci[n]/2 - LucasL[n]/2, 2^(n - 2) + (n + 1)/2 - Binomial[n - 1, (n - 1)/2]/2 - Fibonacci[n - 3] - 3*Fibonacci[n]/2 + LucasL[n]/2], {n, 1, 40}] (* _Vaclav Kotesovec_, Jun 20 2021 *)
%o A325983 (GAP) List([1..40], n->Sum([0..Int((n-1)/2)], k->Binomial(n-1, k-1)-Binomial(n-k-1, k-1)+1));
%o A325983 (Magma) [(&+[Binomial(n-1, k-1)-Binomial(n-k-1, k-1)+1: k in [0..Floor((n-1)/2)]]): n in [1..40]];
%o A325983 (PARI) a(n) = sum(k=0, floor((n-1)/2), binomial(n - 1, k - 1) - binomial(n - k - 1, k - 1) + 1);
%Y A325983 Cf. A004526, A007318, A325982.
%K A325983 nonn
%O A325983 1,3
%A A325983 _Stefano Spezia_, May 29 2019