A375549 Series expansion of (x - 1)^3/(2*x - 1)^5.

1, 7, 33, 129, 450, 1452, 4424, 12896, 36288, 99200, 264704, 691968, 1777152, 4494336, 11212800, 27639808, 67403776, 162791424, 389742592, 925696000, 2182742016, 5112594432, 11901861888, 27550285824, 63438848000, 145366188032, 331584897024, 753146003456, 1703860502528

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

Formula

Row sums of A375550.

From Vaclav Kotesovec, Sep 23 2024: (Start)

a(n) = Sum_{k=0..n} binomial(n+1, n-k) * hypergeom([4, k-n], [k+2], -1).

a(n) ~ 2^(n-6) * n^4 / 3. (End)

From Andrew Howroyd, Nov 13 2025: (Start)

a(n) = (n + 1)*(n + 6)*(n^2 + 15*n + 32)*2^(n-6)/3.

G.f.: (1 - x)^3/(1 - 2*x)^5. (End)

Maple

gf := (1 - x)^3/(1 - 2*x)^5: ser := series(gf, x, 30):
seq(coeff(ser, x, n), n = 0..28);  # Peter Luschny, Nov 13 2025

Mathematica

Table[Sum[Binomial[n+1, n-k] * HypergeometricPFQ[{4, k-n}, {k+2}, -1], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 23 2024 *)

Prog

(PARI) Vec((1 - x)^3/(1 - 2*x)^5 + O(x^31)) \\ Andrew Howroyd, Nov 13 2025

Crossrefs

Cf. A375550.

Sequence in context: A215054 A350643 A114014 * A229515 A320907 A258458

Adjacent sequences: A375546 A375547 A375548 * A375550 A375551 A375552

Keyword

nonn

Author

Peter Luschny, Sep 23 2024

Extensions

New name using a formula of Andrew Howroyd by Peter Luschny, Nov 13 2025

Status

approved