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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)).

%I A249343 #32 Feb 11 2025 07:15:26
%S A249343 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,
%T A249343 55,42,58,44,30,48,33,18,73,56,39,60,42,24,47,28,9,78,57,36,62,40,18,
%U A249343 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
%N 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)).
%H A249343 Paolo Xausa, <a href="/A249343/b249343.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..6560 from Antti Karttunen).
%H A249343 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 A249343 a(n) = A007949(A001142(n)).
%F A249343 a(n) = Sum_{k=0..n} A243759(n,k).
%F A249343 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
%F A249343 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.
%t A249343 A249343[n_] := Sum[#*((#+1)*3^k - n - 1) & [Floor[n/3^k]], {k, Floor[Log[3, n]]}];
%t A249343 Array[A249343, 100, 0] (* _Paolo Xausa_, Feb 11 2025 *)
%o A249343 (PARI) allocatemem(234567890);
%o A249343 A249343(n) = sum(k=0, n, valuation(binomial(n, k), 3));
%o A249343 for(n=0, 6560, write("b249343.txt", n, " ", A249343(n)));
%o A249343 (Scheme)
%o A249343 (define (A249343 n) (add A243759 (A000217 n) (A000096 n)))
%o A249343 (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))
%o A249343 (Haskell)
%o A249343 a249343 = a007949 . a001142  -- _Reinhard Zumkeller_, Mar 16 2015
%Y A249343 Row sums of A243759.
%Y A249343 Row 2 of array A249421.
%Y A249343 Cf. A001142, A007949, A187059, A249345, A249347.
%K A249343 nonn
%O A249343 0,4
%A A249343 _Antti Karttunen_, Oct 28 2014