A375179 a(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*binomial(2*n, k)^3.

0, 1, 63, 3160, 154175, 7623126, 385867944, 20012582304, 1061140695615, 57352579248250, 3150527165580938, 175455318597459264, 9885323482971453608, 562487110567071566800, 32279596239888192636000, 1866149091367606758188160, 108583784076103277424118335, 6354063452012073380091209250

Offset: 0

Author

Peter Bala, Aug 04 2024

Keywords

• nonn
• easy

Comments

Compare with the identity Sum_{k = 0..n-1} (-1)^(n+k+1)*binomial(2*n, k) = (1/2) * binomial(2*n, n) = (1/2) * A000984(n) for n >= 1.

The central binomial coefficients satisfy the supercongruence (1/2) * binomial(2*p, p) == 1 (mod p^3) for all primes p >= 5 (Wolstenholme's theorem).

We conjecture that for the present sequence the stronger supercongruence a(p) == 1 (mod p^5) holds for all primes p >= 7.

Further, we conjecture that for r >= 2 and prime p >= 5, a(p^r) == a(p^(r-1)) (mod p^(3*r+3)).

More generally, for a positive integer m, define a sequence {b_m(n) : n >= 0} by setting b_m(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*binomial(2*n, k)^(2*m+1). We conjecture that the supercongruence b_m(p) == 1 (mod p^(2*m+3)) holds for all primes p >= 2*m + 5, and for r >= 2, the supercongruence b_m(p^r) == b_m(p^(r-1)) (mod p^(3*r+2*m+1)) also holds for all primes p >= 2*m + 5.

Examples

			Examples of supercongruences:
a(7) - a(1) = 20012582304 - 1 = (7^5)*389*3061 == 0 (mod 7^5);
a(17) - a(1) = 6354063452012073380091209250 - 1 = (17^5)*6007*15725921* 47373254231 == 0 (mod 17^5).
		

Links

• Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], (2011).

Crossrefs

Cf. A000984, A375178, A375180.

Programs

Maple

seq(add( (-1)^(n+k+1)*binomial(2*n, k)^3, k = 0..n-1), n = 0..20);

Formula

a(n) ~ 2^(6*n-1) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Aug 08 2024