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%I #20 Aug 27 2023 03:35:23
%S 1,1,2,33,276,4150,65300,1083425,20965000,399876876,8461219032,
%T 178642861782,4010820554664,90684123972156,2130950905378152,
%U 50560833176021025,1231721051614138800,30294218438009039800,759645100717216142000,19213764100954274616908,493269287121905287769776
%N a(n) = S(5,n), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.
%C 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(5,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 a(n) is the total number of 5-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 H. W. Gould, <a href="http://www.jstor.org/stable/2976965">Problem E2384</a>, Amer. Math. Monthly, 81 (1974), 170-171.
%F a(n) = Sum_{k = 0..floor(n/2)} ( (n - 2*k + 1)/(n - k + 1) * binomial(n,k) )^5.
%F From _Alois P. Heinz_, Apr 02 2023: (Start)
%F a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^5.
%F a(n) = Sum_{j=0..n} A120730(n,j)^5.
%F a(n) = A357824(n,5). (End)
%F a(n) ~ 2^(5*n + 19/2) / (125 * Pi^(5/2) * n^(9/2)). - _Vaclav Kotesovec_, Aug 27 2023
%p seq(add( ( binomial(n,k) - binomial(n,k-1) )^5, k = 0..floor(n/2)), n = 0..20);
%t Table[Sum[(Binomial[n, k] - Binomial[n, k-1])^5, {k,0,Floor[n/2]}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 27 2023 *)
%Y Cf. A003161 ( S(3,n) ), A003162 ( S(3,n)/S(1,n) ), A183069 ( S(3,2*n+1)/ S(1,2*n+1) ), A361888 ( S(5,n)/S(1,n) ), A361889 ( S(5,2*n-1)/S(1,2*n-1) ), A361890 ( S(7,n) ), A361891 ( S(7,n)/S(1,n) ), A361892 ( S(7,2*n-1)/S(1,2*n-1) ).
%Y Column k=5 of A357824.
%Y Cf. A008315, A120730.
%K nonn,easy
%O 0,3
%A _Peter Bala_, Mar 28 2023
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