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

%I A249345 #34 Feb 11 2025 07:15:19
%S A249345 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,
%T A249345 43,36,29,22,15,42,34,26,18,10,41,32,23,14,5,40,30,20,10,0,88,76,64,
%U A249345 52,40,82,69,56,43,30,76,62,48,34,20,70,55,40,25,10,64,48,32,16,0
%N A249345 The exponent of the highest power of 5 dividing the product of the elements on the n-th row of Pascal's triangle.
%H A249345 Paolo Xausa, <a href="/A249345/b249345.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..3124 from Antti Karttunen).
%H A249345 Jeffrey C. Lagarias and Harsh Mehta, <a href="http://arxiv.org/abs/1409.4145">Products of binomial coefficients and unreduced Farey fractions</a>, arXiv:1409.4145 [math.NT], 2014.
%F A249345 a(n) = A112765(A001142(n)).
%F A249345 a(n) = Sum_{k=0..n} A112765(binomial(n,k)).
%F A249345 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
%F A249345 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.
%t A249345 A249345[n_] := Sum[#*((#+1)*5^k - n - 1) & [Floor[n/5^k]], {k, Floor[Log[5, n]]}];
%t A249345 Array[A249345, 100, 0] (* _Paolo Xausa_, Feb 11 2025 *)
%o A249345 (PARI)
%o A249345 allocatemem(234567890);
%o A249345 A249345(n) = sum(k=0, n, valuation(binomial(n, k), 5));
%o A249345 for(n=0, 3124, write("b249345.txt", n, " ", A249345(n)));
%o A249345 (Scheme)
%o A249345 (define (A249345 n) (A112765 (A001142 n)))
%o A249345 (Scheme)
%o A249345 (define (A249345 n) (add (lambda (n) (A112765 (A007318 n))) (A000217 n) (A000096 n)))
%o A249345 (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))
%Y A249345 Row 3 of array A249421.
%Y A249345 Cf. A001142, A007318, A112765, A187059, A249343, A249347.
%K A249345 nonn
%O A249345 0,6
%A A249345 _Antti Karttunen_, Oct 28 2014