experimental-projects/oeis-research/Ralf Stephan/2^A101925(n)/formula.sf

#!/usr/bin/ruby

# a(n) = 2^A101925(n).
# https://oeis.org/A101926

# Formula (Daniel Suteu, Feb 03 2017):
#   a(n) is the reduced numerator of 2^(2*n+1)*(n!)^2/(2*n+1)/(2*n)!.

# Alternate form:
#   a(n) is the reduced numerator of (sqrt(π) Γ(n + 1))/Γ(n + 3/2).

# I came across that formula while studying some limits for Pi, related to Wallis product (Product_{k=1..n} (4*k^2)/(4*k^2-1) = Pi/2). (see: A002454)

# In particular:
#     A002454(n) / (2*n+1)! = A046161(n) / A001803(n)
#   2*A002454(n) / (2*n+1)! = A101926(n) / A001803(n)
#   Lim_{n->oo} n*A002454(n) / ((2n+1)!!)^2 = Pi/4
#   Lim_{n->oo} A002454(n) / A079484(n) = Pi/2

# Since A101926(n) is even and A001803(n) is odd, we get:
#   A101926(n) = numerator(2*A002454(n) / (2*n+1)!)
#   A101926(n) = 2 * A046161(n)

# Where:
#   A046161(n) = 2^A005187(n)          -- see "formula" section
#   A101926(n) = 2^(1 + A005187(n))    -- by definition

# which proves the identity:
#   A101926(n) = 2 * A046161(n)

# This program checks the formula against the definition for the first 10^3 terms.

func A005187(n) {
    2*n - popcount(2*n)
}

func A101925(n) {
    A005187(n) + 1
}

func A101926(n) {
    2**A101925(n)
}

for n in (1..1e3) {
    var t = (2**(2*n + 1) * (n!)**2 / (2*n + 1) / (2*n)!)
    var num = t.nu

    say [n, num]
    assert_eq(num, A101926(n))
}