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A362728
a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} A108628(k-1)*x^k/k ).
0
1, 1, 9, 91, 985, 11101, 128475, 1515032, 18116825, 218988046, 2669804209, 32776883899, 404733925435, 5022161428571, 62578069656776, 782560813918216, 9817011145746649, 123492956278927438, 1557295053170126994, 19681186581532094418
OFFSET
0,3
COMMENTS
A108628(n) = B(n+1,n,n+1) in the notation of Straub, equation 24, where it is shown that the supercongruences A108628(n*p^k) == A108628(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and all positive integers n and k.
Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = A108628(n) and, for i >= 1, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. In this notation the present sequence is {a(1,n)}.
We conjecture that the sequences {a(i,n) : n >= 0}, i >= 1, satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7, and positive integers n and r.
LINKS
Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.
FORMULA
Conjecture: the supercongruence a(n*p^r) == a(n(p^(r-1)) (mod p^(3*r)) holds for
all primes p >= 7 and positive integers n and r.
MAPLE
A108628 := proc(n) add(binomial(n, k)*binomial(n+1, k)*binomial(n+k+1, k), k = 0..n) end:
E(n, x) := series( exp(n*add(A108628(k-1)*x^k/k, k = 1..20)), x, 21 ):
seq(coeftayl(E(n, x), x = 0, n), n = 0..20);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, May 03 2023
STATUS
approved