A034256 Expansion of 2 - (1 - 16*x)^(1/4), related to quartic factorial numbers A034176.
1, 4, 24, 224, 2464, 29568, 374528, 4922368, 66451968, 915560448, 12817846272, 181780365312, 2605518569472, 37679807004672, 549048616353792, 8052713039855616, 118777517337870336, 1760702021714313216, 26214896767746441216, 391843720107367858176, 5877655801610517872640
Offset: 0
Programs
-
Mathematica
a[n_] := 4^(2*n-1)*Pochhammer[3/4, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Aug 19 2025 *)
Formula
Equals 4 * A025749(n), n > 0.
a(n) = 4^n*3*A034176(n-1)/n!, n >= 2, where 3*A034176(n-1) = (4*n-5)(!^4) = Product_{j=2..n} (4*j - 5).
O.g.f.: A(x) = 2 - (1 - 16*x)^(1/4).
From Peter Bala, Nov 19 2015: (Start)
For n >= 1, a(n) = (1/(sqrt(2)*Pi)) * Integral_{x = 0..16} x^(n-1)*((16 - x)/x)^(1/4) dx.
It appears that sqrt(A(x)) = 1 + 2*x + 10*x^2 + 92*x^3 + 998*x^4 + 11868*x^5 + 149316*x^6 + ... has integer coefficients. (End)
a(n) ~ 4^(2*n-1) * n^(-5/4) / Gamma(3/4). - Amiram Eldar, Aug 19 2025
Extensions
More terms from Amiram Eldar, Aug 19 2025