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A361891 a(n) = S(7,n)/S(1,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

%I A361891 #24 Mar 25 2025 13:06:17
%S A361891 1,1,1,43,386,9451,246961,6031627,212559508,6571985126,243940325734,
%T A361891 9140730357409,352312505157354,14801600281919487,600054439936968241,
%U A361891 26927918031565051915,1149140935414286560040,53804800109969394477580,2401141625752684697505820
%N A361891 a(n) = S(7,n)/S(1,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.
%C A361891 For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. Gould (1974) conjectured that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n). The present sequence is {S(7,n)/S(1,n)}.
%H A361891 Seiichi Manyama, <a href="/A361891/b361891.txt">Table of n, a(n) for n = 0..563</a>
%H A361891 H. W. Gould, <a href="http://www.jstor.org/stable/2976965">Problem E2384</a>, Amer. Math. Monthly, 81 (1974), 170-171.
%F A361891 a(n) = 1/binomial(n,floor(n/2)) * Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^7.
%F A361891 a(n) ~ 3 * 2^(6*n+13) / (2401 * Pi^3 * n^6). - _Vaclav Kotesovec_, Mar 24 2025
%p A361891 seq(add( ( binomial(n,k) - binomial(n,k-1) )^7/binomial(n,floor(n/2)), k = 0..floor(n/2)), n = 0..20);
%t A361891 Table[Sum[(Binomial[n, k]-Binomial[n, k-1])^7/Binomial[n, Floor[n/2]], {k, 0, Floor[n/2]}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 24 2025 *)
%o A361891 (PARI) s(r, n) = sum(k=0, n\2, (binomial(n, k)-binomial(n, k-1))^r);
%o A361891 a(n) = s(7, n)/s(1, n); \\ _Seiichi Manyama_, Mar 24 2025
%o A361891 (Python)
%o A361891 from math import comb
%o A361891 def A361891(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**7 for j in range((n>>1)+1))//comb(n,n>>1) # _Chai Wah Wu_, Mar 25 2025
%Y A361891 Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A183069 ( S(3,2*n-1)/ S(1,2*n-1) ), A361887 ( S(5,n) ), A361888( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361890 ( S(7,n) ), A361892 ( S(7,2*n-1)/S(1,2*n-1) ).
%K A361891 nonn,easy
%O A361891 0,4
%A A361891 _Peter Bala_, Mar 30 2023