A249343 - OEIS

A249343

The exponent of the highest power of 3 dividing the product of the elements on the n-th row of Pascal's triangle (A001142(n)).

0, 0, 0, 2, 1, 0, 4, 2, 0, 14, 10, 6, 13, 8, 3, 12, 6, 0, 28, 20, 12, 24, 15, 6, 20, 10, 0, 68, 55, 42, 58, 44, 30, 48, 33, 18, 73, 56, 39, 60, 42, 24, 47, 28, 9, 78, 57, 36, 62, 40, 18, 46, 23, 0, 136, 110, 84, 114, 87, 60, 92, 64, 36, 132, 102, 72, 107, 76, 45, 82, 50, 18, 128, 94, 60, 100, 65, 30, 72, 36, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..6560 from Antti Karttunen).

Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

FORMULA

a(n) = A007949(A001142(n)).

a(n) = Sum_{k=0..n} A243759(n,k).

a(n) = Sum_{i=1..n} (2*i-n-1)*v_3(i), where v_3(i) = A007949(i) is the exponent of the highest power of 3 dividing i. - Ridouane Oudra, Jun 02 2022

a(n) = Sum_{k=1..floor(log_3(n))} t*((t+1)*3^k - n - 1), where t = floor(n/(3^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.

MATHEMATICA

A249343[n_] := Sum[#*((#+1)*3^k - n - 1) & [Floor[n/3^k]], {k, Floor[Log[3, n]]}];

Array[A249343, 100, 0] (* Paolo Xausa, Feb 11 2025 *)

PROG

(PARI) allocatemem(234567890);

A249343(n) = sum(k=0, n, valuation(binomial(n, k), 3));

for(n=0, 6560, write("b249343.txt", n, " ", A249343(n)));

(Scheme)

(define (A249343 n) (add A243759 (A000217 n) (A000096 n)))

(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))

(Haskell)

a249343 = a007949 . a001142 -- Reinhard Zumkeller, Mar 16 2015