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

1,6

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.

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

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 *)

Prog

(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

Crossrefs

Cf. A000120, A005130, A007814, A227833.

Sequence in context: A237619 A156747 A318958 * A335359 A332205 A219237

Adjacent sequences: A194824 A194825 A194826 * A194828 A194829 A194830

Keyword

nonn,easy

Author

R. J. Mathar, Sep 03 2011

Status

approved