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A364116
a(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^n for n >= 0.
6
1, 3, 73, 5623, 908001, 251831261, 106898093065, 64439674636863, 52344140654486017, 55113399257643294769, 73004404532578627776801, 118810038754810358401521065, 233027150139808176596750408337, 542098915811219991386976197616441
OFFSET
0,2
COMMENTS
Main diagonal of A364113.
Compare with the two types of Apéry numbers A005258 and A005259, which are related to the Legendre polynomials by A005258(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x)) and A005259(n) = [x^k] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^2.
A005258 is the main diagonal of A108625 and A005259 is the main diagonal of A143007.
FORMULA
Conjectures:
1) a(p) == 2*p + 1 (mod p^4) for all primes p >= 3 (checked up to p = 101).
More generally, the supercongruence a(p^k) == 2*p^k + 1 (mod p^(3+k)) may hold for all primes p >= 5 and all k >= 1.
2) a(p-1) == 1 (mod p^3) for all primes p >= 5 (checked up to p = 101).
More generally, the supercongruence a(p^k - p^(k-1)) == 1 (mod p^(2+k)) may hold for all primes p >= 5 and all k >= 1.
From Vaclav Kotesovec, Jul 10 2023: (Start)
a(n) ~ c * d^n * n^(2*n - 1/2), where d = 2.102423770105721036432437141524634595160013830317976222331887376263238499... (the same as for A033935) and c = 1.325068544739430738025458046917491360304162175529817456184402029433873399...
a(n) ~ A033935(n) * exp(2*n + 1) / (2*Pi*n).
a(n) ~ A033935(n) * exp(1) * n^(2*n) / n!^2. (End)
MAPLE
a(n) := coeff(series( 1/(1-x)* LegendreP(n, (1+x)/(1-x))^n, x, 21), x, n):
seq(a(n), n = 0..20);
MATHEMATICA
Table[SeriesCoefficient[1/(1 - x) * LegendreP[n, (1 + x)/(1 - x)]^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 09 2023 *)
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Jul 08 2023
STATUS
approved