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A249347 The exponent of the highest power of 7 dividing the product of the elements on the n-th row of Pascal's triangle.

%I A249347 #34 Feb 11 2025 03:14:51
%S A249347 0,0,0,0,0,0,0,6,5,4,3,2,1,0,12,10,8,6,4,2,0,18,15,12,9,6,3,0,24,20,
%T A249347 16,12,8,4,0,30,25,20,15,10,5,0,36,30,24,18,12,6,0,90,82,74,66,58,50,
%U A249347 42,89,80,71,62,53,44,35,88,78,68,58,48,38,28,87,76,65,54,43,32,21,86,74,62,50,38,26,14,85,72,59,46,33,20,7,84,70,56,42,28,14,0
%N A249347 The exponent of the highest power of 7 dividing the product of the elements on the n-th row of Pascal's triangle.
%H A249347 Paolo Xausa, <a href="/A249347/b249347.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2400 from Antti Karttunen).
%H A249347 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 A249347 a(n) = A214411(A001142(n)).
%F A249347 a(n) = Sum_{k=0..n} A214411(binomial(n,k)).
%F A249347 a(n) = Sum_{i=1..n} (2*i-n-1)*v_7(i), where v_7(i) = A214411(i) is the exponent of the highest power of 7 dividing i. - _Ridouane Oudra_, Jun 03 2022
%F A249347 a(n) = Sum_{k=1..floor(log_7(n))} t*((t+1)*7^k - n - 1), where t = floor(n/(7^k)). - _Paolo Xausa_, Feb 09 2025, derived from _Ridouane Oudra_'s formula above.
%t A249347 A249347[n_] := Sum[#*((#+1)*7^k - n - 1) & [Floor[n/7^k]], {k, Floor[Log[7, n]]}];
%t A249347 Array[A249347, 100, 0] (* _Paolo Xausa_, Feb 08 2025 *)
%o A249347 (PARI)
%o A249347 allocatemem(234567890);
%o A249347 A249347(n) = sum(k=0, n, valuation(binomial(n, k), 7));
%o A249347 for(n=0, 2400, write("b249347.txt", n, " ", A249347(n)));
%o A249347 (Scheme)
%o A249347 (define (A249347 n) (A214411 (A001142 n)))
%o A249347 (Scheme)
%o A249347 (define (A249347 n) (add (lambda (n) (A214411 (A007318 n))) (A000217 n) (A000096 n)))
%o A249347 (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))
%Y A249347 Row 4 of array A249421.
%Y A249347 Cf. A001142, A007318, A214411, A187059, A249343, A249345.
%K A249347 nonn
%O A249347 0,8
%A A249347 _Antti Karttunen_, Oct 28 2014