A004981 -id:A004981 - cp's OEIS frontend

A376835 Expansion of 1/((1-x)^4 - 8*x^4)^(1/4).

1, 1, 1, 1, 3, 11, 31, 71, 151, 343, 871, 2311, 6001, 15081, 37493, 94381, 241931, 625771, 1617211, 4164763, 10719793, 27674473, 71722773, 186353453, 484657729, 1260984161, 3283294561, 8559401761, 22343836711, 58391858383, 152722920691, 399719304411

Offset: 0

Seiichi Manyama, Oct 06 2024

nonn

Cf. A002426, A376836.

Cf. A004981.

f:= 1/((1-x)^4 - 8*x^4)^(1/4):
S:= series(f,x,41):
seq(coeff(S,x,i),i=0..40); # Robert Israel, Oct 06 2024

a[n_]:=Sum[(-8)^k * Binomial[-1/4,k] * Binomial[n,n-4*k],{k,0,Floor[n/4]}]; Array[a,32,0] (* Stefano Spezia, Oct 06 2024 *)

my(N=40, x='x+O('x^N)); Vec(1/((1-x)^4-8*x^4)^(1/4))

a(n) = Sum_{k=0..floor(n/4)} (-8)^k * binomial(-1/4,k) * binomial(n,n-4*k).

(7 + 7*n)*a(n) + (7 + 4*n)*a(n + 1) - (15 + 6*n)*a(n + 2) + (13 + 4*n)*a(n + 3) - (n + 4)*a(n + 4) = 0. - Robert Israel, Oct 06 2024

A224882 G.f.: 1/(1 - 32*x)^(1/16).

Original entry on oeis.org

1, 2, 34, 748, 18326, 476476, 12864852, 356540184, 10072260198, 288738125676, 8373405644604, 245112419778408, 7230816383463036, 214699624924363992, 6410317372741724904, 192309521182251747120, 5793324325615333881990, 175162864903898918549580

Offset: 0

Paul D. Hanna, Jul 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 34*x^2 + 748*x^3 + 18326*x^4 + 476476*x^5 +...
where
A(x)^16 = 1 + 32*x + 1024*x^2 + 32768*x^3 + 1048576*x^4 +...+ 32^n*x^n +...
Also,
A(x)^8 = 1 + 16*x + 384*x^2 + 10240*x^3 + 286720*x^4 +...+ 8^n*A000984(n)*x^n +...
A(x)^4 = 1 + 8*x + 160*x^2 + 3840*x^3 + 99840*x^4 +...+ 4^n*A004981(n)*x^n +...
A(x)^2 = 1 + 4*x + 72*x^2 + 1632*x^3 + 40800*x^4 +...+ 2^n*A224881(n)*x^n +...

Cf. A224881, A000984, A004981.

CoefficientList[Series[1/(1-32*x)^(1/16), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

{a(n)=polcoeff(1/(1-32*x +x*O(x^n))^(1/16),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=(2^n/n!)*prod(k=0, n-1, 16*k + 1)}
for(n=0, 30, print1(a(n), ", "))

a(n) = (2^n/n!) * Product_{k=0..n-1} (16*k + 1).

a(n) ~ 32^n/(GAMMA(1/16)*n^(15/16)). - Vaclav Kotesovec, Jul 24 2013

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A376835 Expansion of 1/((1-x)^4 - 8*x^4)^(1/4).

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A224882 G.f.: 1/(1 - 32*x)^(1/16).

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