A091904 -id:A091904 - cp's OEIS frontend

A091905 Expansion of (1-4x)/((1+4x)(1-8x)).

1, 0, 32, 128, 1536, 10240, 90112, 688128, 5636096, 44564480, 358612992, 2860515328, 22917677056, 183207198720, 1466194460672, 11727408201728, 93827855548416, 750588484648960, 6004845316145152, 48038212773347328

Offset: 0

Paul Barry, Feb 10 2004

Index entries for linear recurrences with constant coefficients, signature (4,32).

a(n)=8^n/3+2(-4)^n/3 = 32*A091904(n-1).

a(n) = 4^n*A078008(n). - R. J. Mathar, Mar 08 2021

A099138 a(n) = 6^(n-1)*J(n), where J(n) = A001045(n).

Original entry on oeis.org

0, 1, 6, 108, 1080, 14256, 163296, 2006208, 23794560, 287214336, 3436494336, 41298398208, 495217981440, 5944792559616, 71324450021376, 855971764420608, 10271190988062720, 123257112966660096, 1479068428940476416

Offset: 0

Paul Barry, Sep 29 2004

In general k^(n-1)*J(n), where J(n) = A001045(n), is given by ((2*k)^n - (-k)^n)/(3*k) with g.f. x/((1+k*x)*(1-2*k*x)).

G. C. Greubel, Table of n, a(n) for n = 0..925
Index entries for linear recurrences with constant coefficients, signature (6,72).

Cf. A001045, A003683, A080424, A091903, A091904.

[(12^n - (-6)^n)/18: n in [0..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{6,72}, {0,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

[(12^n - (-6)^n)/18 for n in range(41)] # G. C. Greubel, Feb 18 2023

G.f.: x/((1+6*x)*(1-12*x)).
a(n) = 6^(n-1)*Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k) * 2^k.
a(n) = (12^n - (-6)^n)/18.
a(n) = 6^(n-1)*A001045(n).
E.g.f.: (1/18)*(exp(12*x) - exp(-6*x)). - G. C. Greubel, Feb 18 2023

A159277 Ways to write the identity as a product of n 3-cycles in symmetric group S_4.

Original entry on oeis.org

1, 0, 8, 32, 384, 2560, 22528, 172032, 1409024, 11141120, 89653248, 715128832, 5729419264, 45801799680, 366548615168, 2931852050432, 23456963887104, 187647121162240, 1501211329036288, 12009553193336832, 96076975302508544

Offset: 0

Jacob A. Siehler, Apr 07 2009

Flavien Mabilat, Some counting formulas for λ-quiddities over the rings Z / 2^m Z, arXiv:2402.09968 [math.CO], 2024.

Cf. A091904.

a(n+1) = (2/3)*(-1)^n*((-8)^n-4^n).
O.g.f.: 1 - 8*x^2/(32*x^2+4*x-1).
a(n) = 8 * A091904(n-1). - R. J. Mathar, Jun 28 2009

Offset corrected by R. J. Mathar, Jun 28 2009

Offset changed back and a(0) = 1 prepended by Andrey Zabolotskiy, Feb 21 2024

cp's OEIS Frontend

A091905 Expansion of (1-4x)/((1+4x)(1-8x)).

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A099138 a(n) = 6^(n-1)*J(n), where J(n) = A001045(n).

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A159277 Ways to write the identity as a product of n 3-cycles in symmetric group S_4.

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