A034256 - OEIS

%I #33 Apr 13 2026 11:36:33

%S 1,4,24,224,2464,29568,374528,4922368,66451968,915560448,12817846272,

%T 181780365312,2605518569472,37679807004672,549048616353792,

%U 8052713039855616,118777517337870336,1760702021714313216,26214896767746441216,391843720107367858176,5877655801610517872640

%N Expansion of 2 - (1 - 16*x)^(1/4), related to quartic factorial numbers A034176.

%H Harvey P. Dale, <a href="/A034256/b034256.txt">Table of n, a(n) for n = 0..834</a>

%F a(n) = 4 * A025749(n), n > 0.

%F 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).

%F O.g.f.: A(x) = 2 - (1 - 16*x)^(1/4).

%F From _Peter Bala_, Nov 19 2015: (Start)

%F For n >= 1, a(n) = (1/(sqrt(2)*Pi)) * Integral_{x = 0..16} x^(n-1)*((16 - x)/x)^(1/4) dx.

%F 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)

%F a(n) ~ 4^(2*n-1) * n^(-5/4) / Gamma(3/4). - _Amiram Eldar_, Aug 19 2025

%F Sum_{n>=0} 1/a(n) = (58/45) + (2*sqrt(2)/(3*15^(5/4)))*(2*arcsin(15^(1/4)/(2*sqrt(2))) - arcsinh(sqrt(15+8*sqrt(15))/4)). - _Amiram Eldar_, Dec 22 2025

%t a[n_] := 4^(2*n-1)*Pochhammer[3/4, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* _Amiram Eldar_, Aug 19 2025 *)

%t CoefficientList[Series[2-Surd[1-16x,4],{x,0,20}],x] (* _Harvey P. Dale_, Apr 13 2026 *)

%Y Cf. A025749, A034176, A068465.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_

%E More terms from _Amiram Eldar_, Aug 19 2025