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A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5*n-2*k-1,n-2*k).

%I A371753 #31 Aug 16 2025 10:25:10
%S A371753 1,4,37,376,4013,44064,492871,5585080,63901421,736575316,8540549322,
%T A371753 99503540008,1163910870767,13660217796736,160782910480936,
%U A371753 1897131524755896,22433316399634669,265775992115557076,3154067508987675679,37487016824453703920,446148092364247390618
%N A371753 a(n) = Sum_{k=0..floor(n/2)} binomial(5*n-2*k-1,n-2*k).
%F A371753 a(n) = [x^n] 1/((1-x^2) * (1-x)^(4*n)).
%F A371753 a(n) ~ 5^(5*n + 3/2) / (3 * sqrt(Pi*n) * 2^(8*n + 5/2)). - _Vaclav Kotesovec_, Apr 05 2024
%F A371753 Conjecture D-finite with recurrence +1024*n*(796184150374453*n -1374782084855770) *(4*n-3)*(2*n-1)*(4*n-1)*a(n) +64*(-4720591427354845074*n^5 +16046598674673412696*n^4 -14164434258362644374*n^3 -6132680339747354209*n^2 +16406971563067867560*n -7312237120275595200)*a(n-1) +40*(-4968388566264801507*n^5 +51044954667717039608*n^4 -218029351288077225930*n^3 +471970442274586326109*n^2 -511707487331990011785*n +221366817798624198360)*a(n-2) -25*(5*n-11) *(719005061479699*n -1438086256867727)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - _R. J. Mathar_, Sep 27 2024
%F A371753 From _Seiichi Manyama_, Aug 05 2025: (Start)
%F A371753 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+1,k).
%F A371753 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*n+k,k). (End)
%F A371753 From _Seiichi Manyama_, Aug 14 2025: (Start)
%F A371753 a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(5*n+1,k) * binomial(5*n-k,n-k).
%F A371753 G.f.: g^2/((-1+2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. (End)
%F A371753 G.f.: B(x)^2/(1 + 6*(B(x)-1)/5), where B(x) is the g.f. of A001449. - _Seiichi Manyama_, Aug 15 2025
%F A371753 G.f.: 1/(1 - x*g^3*(-5+9*g)) where g = 1+x*g^5 is the g.f. of A002294. - _Seiichi Manyama_, Aug 16 2025
%p A371753 A371753 := proc(n)
%p A371753     add( binomial(5*n-2*k-1,n-2*k),k=0..floor(n/2)) ;
%p A371753 end proc:
%p A371753 seq(A371753(n),n=0..50) ; # _R. J. Mathar_, Sep 27 2024
%o A371753 (PARI) a(n) = sum(k=0, n\2, binomial(5*n-2*k-1, n-2*k));
%Y A371753 Cf. A026641, A147855, A183160.
%Y A371753 Cf. A001449, A079589, A079678, A385632, A386812.
%K A371753 nonn
%O A371753 0,2
%A A371753 _Seiichi Manyama_, Apr 05 2024