A249345 The exponent of the highest power of 5 dividing the product of the elements on the n-th row of Pascal's triangle.

0, 0, 0, 0, 0, 4, 3, 2, 1, 0, 8, 6, 4, 2, 0, 12, 9, 6, 3, 0, 16, 12, 8, 4, 0, 44, 38, 32, 26, 20, 43, 36, 29, 22, 15, 42, 34, 26, 18, 10, 41, 32, 23, 14, 5, 40, 30, 20, 10, 0, 88, 76, 64, 52, 40, 82, 69, 56, 43, 30, 76, 62, 48, 34, 20, 70, 55, 40, 25, 10, 64, 48, 32, 16, 0

Offset: 0

Author

Antti Karttunen, Oct 28 2014

Keywords

• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..3124 from Antti Karttunen).
• Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Crossrefs

Row 3 of array A249421.

Cf. A001142, A007318, A112765, A187059, A249343, A249347.

Programs

Mathematica

A249345[n_] := Sum[#*((#+1)*5^k - n - 1) & [Floor[n/5^k]], {k, Floor[Log[5, n]]}];
Array[A249345, 100, 0] (* Paolo Xausa, Feb 11 2025 *)

PARI

allocatemem(234567890);
A249345(n) = sum(k=0, n, valuation(binomial(n, k), 5));
for(n=0, 3124, write("b249345.txt", n, " ", A249345(n)));

Scheme

(define (A249345 n) (A112765 (A001142 n)))

Scheme

(define (A249345 n) (add (lambda (n) (A112765 (A007318 n))) (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)))))))

Formula

a(n) = A112765(A001142(n)).

a(n) = Sum_{k=0..n} A112765(binomial(n,k)).

a(n) = Sum_{i=1..n} (2*i-n-1)*v_5(i), where v_5(i) = A112765(i) is the exponent of the highest power of 5 dividing i. - Ridouane Oudra, Jun 02 2022

a(n) = Sum_{k=1..floor(log_5(n))} t*((t+1)*5^k - n - 1), where t = floor(n/(5^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.