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A194827 2-adic valuation of the number of n X n Alternating Sign Matrices (A005130(n)). 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 14 16:43 EDT 2022. Contains 356122 sequences. (Running on oeis4.)