A362730 a(n) = [x^n] E(x)^n where E(x) = exp( Sum_{k >= 1} binomial(2*k,k)^2*x^k/k ).

1, 4, 68, 1336, 27972, 607004, 13478072, 304083224, 6941422916, 159882680452, 3708781743068, 86526900550864, 2028273983776440, 47733938489878528, 1127187050415921304, 26694934151138897336, 633813403549444601156, 15082008687681962081088, 359592614152718921447108

Offset

0,2

Comments

Compare with A359108(n) = [x^n] F(x)^n where F(x) = exp( Sum_{k >= 1} binomial(2*k,k)*x^k/k ).

More generally, we inductively define a family of sequences {a(i,n) : n >= 0}, i >= 0, by setting a(0,n) = binomial(2*n,n)^2 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 >= 5, and positive integers n and r.

Links

Table of n, a(n) for n=0..18.

Formula

Conjecture: the supercongruence a(n*p^r) == a(n(p^(r-1)) (mod p^(3*r)) holds for all primes p >= 5 and positive integers n and r.

Maple

E(n, x) := series( exp(n*add(binomial(2*k, k)^2*x^k/k, k = 1..20)), x, 21 ):
seq(coeftayl(E(n, x), x = 0, n), n = 0..20);

Crossrefs

Cf. A000984, A002894, A359108, A362722 - A362733.

Sequence in context: A323276 A141032 A156084 * A000658 A351027 A156470

Adjacent sequences: A362727 A362728 A362729 * A362731 A362732 A362733

Keyword

nonn,easy

Author

Peter Bala, May 05 2023

Status

approved