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A386371 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n+1,k).

%I A386371 #30 Aug 31 2025 12:50:00
%S A386371 1,3,31,317,3399,37418,419229,4756104,54463335,628197809,7287712566,
%T A386371 84942987198,993941174829,11668806723876,137378189197112,
%U A386371 1621322803014672,19175540677541991,227217662222902443,2696878158795639549,32057403690640189635,381573145993865438254
%N A386371 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n+1,k).
%H A386371 Vincenzo Librandi, <a href="/A386371/b386371.txt">Table of n, a(n) for n = 0..800</a>
%F A386371 a(n) = [x^n] (1+x)^(5*n+1)/(1+3*x).
%F A386371 a(n) = [x^n] 1/((1-x)^(4*n+1) * (1+2*x)).
%F A386371 a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
%F A386371 a(n) = Sum_{k=0..n} (-2)^k * binomial(5*n-k,n-k).
%F A386371 G.f.: 1/(1 - x*g^3*(-10+13*g)) where g = 1+x*g^5 is the g.f. of A002294.
%F A386371 G.f.: g^2/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294.
%F A386371 G.f.: B(x)^2/(1 + 7*(B(x)-1)/5), where B(x) is the g.f. of A001449.
%F A386371 D-finite with recurrence 648*n*(135551509682187347695*n -244103380745409504343) *(4*n-1)*(2*n-1)*(4*n-3)*a(n) +(-33979500619583537984836075*n^5 +130803893690808003041848009*n^4 -168380151442376797602371231*n^3 +62069291513227826684567999*n^2 +49760069127090078338544954*n -39530305857276050670355320)*a(n-1) +40*(-108999332467309598098777*n^5 -28981701912184019189355*n^4 -1554974299825191814369159*n^3 +13581461461293413639358363*n^2 -28599284433109723900055776*n +18909354537435947334628944)*a(n-2) +211200*(5*n-11) *(5*n-9)*(28440609019752807*n +93502568692163852)*(5*n-13)*(5*n-12)*a(n-3)=0. - _R. J. Mathar_, Aug 26 2025
%t A386371 Table[Sum[(-3)^(n-k)*Binomial[5*n+1,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 31 2025 *)
%o A386371 (PARI) a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n+1, k));
%o A386371 (Magma) [&+[(-3)^(n-k) * Binomial(5*n+1,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 31 2025
%Y A386371 Cf. A001449, A079589, A079678, A371753, A385632, A386812.
%Y A386371 Cf. A226705, A226733, A226751, A387085.
%Y A386371 Cf. A002294.
%K A386371 nonn,changed
%O A386371 0,2
%A A386371 _Seiichi Manyama_, Aug 17 2025