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

0,2

Links

Harvey P. Dale, Table of n, a(n) for n = 0..834

Formula

a(n) = 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

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

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

Crossrefs

Cf. A025749, A034176, A068465.

Sequence in context: A334602 A183279 A145237 * A179539 A277612 A216857

Adjacent sequences: A034253 A034254 A034255 * A034257 A034258 A034259

Keyword

easy,nonn

Author

Wolfdieter Lang

Extensions

More terms from Amiram Eldar, Aug 19 2025

Status

approved