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

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, 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, 18, 18, 20, 20, 21, 20, 20, 18, 18, 16, 17, 16, 16, 14, 13, 10, 11, 10, 11, 10, 10

Offset: 1

Author

R. J. Mathar, Sep 03 2011

Keywords

• nonn
• easy

Links

• Kenny Lau, Table of n, a(n) for n = 1..9999
• Clemens Heuberger and Helmut Prodinger, A precise description of the p-adic valuation of the number of alternating sign matrices, Intl. J. Numb. Th., Vol. 7, No. 1 (2011), pp. 57-69.

Crossrefs

Cf. A000120, A005130, A007814, A227833.

Programs

Maple

Sp := proc(n,p) add(d,d=convert(n,base,p)) ; end proc:
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:
A194827 := proc(n) nuA005130(n,2) ; end proc:

Mathematica

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 *)

Python

# a(n) = prod(k=0, n-1, (3k+1)!/(n+k)!)
# a(n+1) = prod(k=0, n, (3k+1)!/(n+k+1)!)
# a(n+1) = prod(k=0, n, (3k+1)!/(n+k)!) prod(k=0, n, 1/(n+k+1))
# a(n+1)/a(n) = [(3n+1)!/(2n)!] [n!/(2n+1)!]
n=10000; N=3*n+1; val=[0]*(N+1); exp=2
while exp <= N:
    for j in range(exp,N+1,exp): val[j] += 1
    exp *= 2
fac_val=[0]*(N+1)
for i in range(N): fac_val[i+1] = fac_val[i] + val[i+1]
res=0
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]
# Kenny Lau, Jun 09 2018

Formula

a(n) = A007814(A005130(n)).

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