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A194827 2-adic valuation of the number of n X n Alternating Sign Matrices (A005130(n)).

%I A194827 #29 Dec 16 2024 22:19:20
%S A194827 0,1,0,1,0,2,2,3,2,2,0,2,2,4,4,5,4,4,2,2,0,3,4,6,6,7,6,8,8,10,10,11,
%T A194827 10,10,8,8,6,7,6,6,4,3,0,3,4,7,8,10,10,11,10,11,10,13,14,16,16,17,16,
%U A194827 18,18,20,20,21,20,20,18,18,16,17,16,16,14,13,10,11,10,11,10,10
%N A194827 2-adic valuation of the number of n X n Alternating Sign Matrices (A005130(n)).
%H A194827 Kenny Lau, <a href="/A194827/b194827.txt">Table of n, a(n) for n = 1..9999</a>
%H A194827 Clemens Heuberger and Helmut Prodinger, <a href="http://dx.doi.org/10.1142/S1793042111003892">A precise description of the p-adic valuation of the number of alternating sign matrices</a>, Intl. J. Numb. Th., Vol. 7, No. 1 (2011), pp. 57-69.
%F A194827 a(n) = A007814(A005130(n)).
%F A194827 a(n) = a(n-1) + s(2*n-2) + s(2*n-1) - s(n-1) - s(3*n-2), where s(n) = A000120(n). - _Amiram Eldar_, Feb 21 2021
%p A194827 Sp := proc(n,p) add(d,d=convert(n,base,p)) ; end proc:
%p A194827 nuA005130 := proc(n,p) add(Sp(n+j,p),j=0..n-1)-add(Sp(3*j+1,p),j=0..n-1) ; %/(p-1) ; end proc:
%p A194827 A194827 := proc(n) nuA005130(n,2) ; end proc:
%t A194827 s[n_] := DigitCount[n, 2, 1]; a[0] = 0; a[n_] := a[n] = a[n - 1] + s[2*n - 2] + s[2*n - 1] - s[n - 1] - s[3*n - 2]; Array[a, 100] (* _Amiram Eldar_, Feb 21 2021 *)
%o A194827 (Python)
%o A194827 # a(n) = prod(k=0, n-1, (3k+1)!/(n+k)!)
%o A194827 # a(n+1) = prod(k=0, n, (3k+1)!/(n+k+1)!)
%o A194827 # a(n+1) = prod(k=0, n, (3k+1)!/(n+k)!) prod(k=0, n, 1/(n+k+1))
%o A194827 # a(n+1)/a(n) = [(3n+1)!/(2n)!] [n!/(2n+1)!]
%o A194827 n=10000; N=3*n+1; val=[0]*(N+1); exp=2
%o A194827 while exp <= N:
%o A194827     for j in range(exp,N+1,exp): val[j] += 1
%o A194827     exp *= 2
%o A194827 fac_val=[0]*(N+1)
%o A194827 for i in range(N): fac_val[i+1] = fac_val[i] + val[i+1]
%o A194827 res=0
%o A194827 for i in range(1,n): print(i,res); res += fac_val[3*i+1] + fac_val[i] - fac_val[2*i] - fac_val[2*i+1]
%o A194827 # _Kenny Lau_, Jun 09 2018
%Y A194827 Cf. A000120, A005130, A007814, A227833.
%K A194827 nonn,easy
%O A194827 1,6
%A A194827 _R. J. Mathar_, Sep 03 2011