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A181990 a(n) = Sum_{0 <= k <= m < p} (binomial(m, k)^(p-1))/p, where p is the n-th prime. 0
3, 399, 12708885, 124515078454872901983423, 39212583445587381894247266262023061, 43487633454143579523135045521112077473364484383507327790688372131, 157851796824901989964381293031623545741924564754192453966085327785455257503133278729 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

a(n) is a sum of all elements in the first p rows of Pascal's triangle each raised to the (p-1) power and divided by p, where p is the n-th prime.

For p = 3 and 7 (and their powers like 3, 9, 27, ... and 7, 49, ...) the sums of all elements in n = p^k top rows of Pascal's triangle each raised to the (n-1) = (p^k-1) power are divisible by n^2 = p^(2k) for all k > 0.

LINKS

Table of n, a(n) for n=2..8.

Eric Weisstein's World of Mathematics, Pascal's Triangle

Eric Weisstein's World of Mathematics, Binomial Sums

MATHEMATICA

Table[(Sum[Binomial[m, k]^(Prime[n] - 1), {m, 0, Prime[n] - 1}, {k, 0, m}])/Prime[n], {n, 2, 10}]

PROG

(PARI) a(n) = my(p=prime(n)); sum(m=0, p-1, sum(k=0, m, binomial(m, k)^(p-1))/p); \\ Michel Marcus, Dec 03 2018

CROSSREFS

Cf. A007318, A006134, A083096, A066796, A083097, A081601, A010060, A122485, A167912.

Sequence in context: A199146 A137023 A203563 * A198666 A330014 A305664

Adjacent sequences:  A181987 A181988 A181989 * A181991 A181992 A181993

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Apr 04 2012

STATUS

approved

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Last modified November 27 16:35 EST 2021. Contains 349394 sequences. (Running on oeis4.)