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A361892
a(n) = S(7,2*n-1)/S(1,2*n-1), where S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.
5
1, 43, 9451, 6031627, 6571985126, 9140730357409, 14801600281919487, 26927918031565051915, 53804800109969394477580, 116002825041515533807200418, 266118189111094898593879923346, 642598035707739308769581970619393
OFFSET
0,2
COMMENTS
Odd bisection of A361891.
Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for positive integers n and r and all primes p >= 5.
FORMULA
a(n) = 1/binomial(2*n-1,n-1) * Sum_{k = 0..n-1} ( (2*n - 2*k)/(2*n - k) * binomial(2*n-1,k) )^7 for n >= 1.
MAPLE
seq(add( ( binomial(2*n-1, k) - binomial(2*n-1, k-1) )^7/binomial(2*n-1, n-1), k = 0..n-1), n = 1..20);
CROSSREFS
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) ), A361891 ( S(7,n)/S(1,n) ).
Sequence in context: A208625 A147522 A183489 * A356203 A184144 A262856
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Mar 30 2023
STATUS
approved