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A361890 a(n) = S(7,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

%I A361890 #25 Mar 25 2025 12:57:51
%S A361890 1,1,2,129,2316,94510,4939220,211106945,14879165560,828070125876,
%T A361890 61472962084968,4223017425122958,325536754765395096,
%U A361890 25399546083773839692,2059386837863675003112,173281152533121109073025,14789443838781868027714800,1307994690673355979749969800
%N A361890 a(n) = S(7,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.
%C A361890 For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(7,n)}. 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).
%C A361890 a(n) is the total number of 7-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2). - _Alois P. Heinz_, Apr 02 2023
%H A361890 Seiichi Manyama, <a href="/A361890/b361890.txt">Table of n, a(n) for n = 0..483</a>
%H A361890 H. W. Gould, <a href="http://www.jstor.org/stable/2976965">Problem E2384</a>, Amer. Math. Monthly, 81 (1974), 170-171.
%F A361890 a(n) = Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^7.
%F A361890 From _Alois P. Heinz_, Apr 02 2023: (Start)
%F A361890 a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^7.
%F A361890 a(n) = Sum_{j=0..n} A120730(n,j)^7.
%F A361890 a(n) = A357824(n,7). (End)
%F A361890 a(n) ~ 3 * 2^(7*n + 27/2) / (2401 * Pi^(7/2) * n^(13/2)). - _Vaclav Kotesovec_, Aug 27 2023
%p A361890 seq(add( ( binomial(n,k) - binomial(n,k-1) )^7, k = 0..floor(n/2)), n = 0..20);
%t A361890 Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^7, {k,0,Floor[n/2]}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 27 2023 *)
%o A361890 (Python)
%o A361890 from math import comb
%o A361890 def A361890(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**7 for j in range((n>>1)+1)) # _Chai Wah Wu_, Mar 25 2025
%Y A361890 Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A382394 ( S(3,2*n-1) ), 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) ), A361891 ( S(7,n)/S(1,n) ), A361892 ( S(7,2*n-1)/ S(1,2*n-1) ).
%Y A361890 Column k=7 of A357824.
%Y A361890 Cf. A008315, A120730.
%K A361890 nonn,easy
%O A361890 0,3
%A A361890 _Peter Bala_, Mar 30 2023