A060195 -id:A060195 - cp's OEIS frontend

A016140 Expansion of 1/((1-3x)(1-8*x)).

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

Offset: 0

N. J. A. Sloane

In general, for expansion of 1/((1-b*x)*(1-c*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - b*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

8*a(n) gives the number of edges in the n-th-order Sierpiński carpet graph. - Eric W. Weisstein, Aug 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

Cf. A190543.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11,-24},{1,11},30] (* Harvey P. Dale, Feb 03 2022 *)

Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,11,24) for n in range(1, 30)] # Zerinvary Lajos, Apr 27 2009

a(n) = (8^(n+1) - 3^(n+1))/5.
a(n) = 11*a(n-1) - 24*a(n-2).
a(n) = 3*a(n-1) + 8^n.
a(n) = 8*a(n-1) + 3^n.
a(n) = Sum_{i=0..n} 3^i*8^(n-i).
E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024

A075503 Stirling2 triangle with scaled diagonals (powers of 8).

Original entry on oeis.org

1, 8, 1, 64, 24, 1, 512, 448, 48, 1, 4096, 7680, 1600, 80, 1, 32768, 126976, 46080, 4160, 120, 1, 262144, 2064384, 1232896, 179200, 8960, 168, 1, 2097152, 33292288, 31653888, 6967296, 537600, 17024, 224, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(8*z) - 1)*x/8) - 1.

			[1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       8        1
*      64       24        1
*     512      448       48       1
*    4096     7680     1600      80      1
*   32768   126976    46080    4160    120     1
*  262144  2064384  1232896  179200   8960   168   1
* 2097152 33292288 31653888 6967296 537600 17024 224 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507.

Cf. A075502, A075504.

Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (8^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*8)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 8m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-8k*x), m >= 1.
E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1.

A065128 Number of invertible n X n matrices mod 4 (i.e., over the ring Z_4).

Original entry on oeis.org

1, 2, 96, 86016, 1321205760, 335522845163520, 1385295986380096143360, 92239345887620476544860815360, 98654363640526679389774053813465907200, 1691558770638735027870457216848672340872423014400, 464518059995994038184379206447729320401459864818351813427200

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 14 2001

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=4 of A316622.

Cf. A048651, A053291, A060195.

a[n_] := 4^(n^2)*Product[1 - 1/2^k, {k, 1, n} ]; Table[ a[n], {n, 0, 10} ]

for(n=1,11,print(4^(n^2)*prod(k=1,n,(1-1/2^k))))

a(n) = 4^(n^2) * Product_{k=1..n} (1 - 1/2^k).
a(n) = 2^(n^2) * A002884(n). - Geoffrey Critzer, Feb 04 2018
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} 2*A060195(k).
a(n) ~ c * 4^(n^2), where c = A048651. (End)

More terms from Robert G. Wilson v and Jason Earls, Nov 16 2001

A076003 Third column of triangle A075503.

Original entry on oeis.org

1, 48, 1600, 46080, 1232896, 31653888, 792985600, 19566428160, 478167433216, 11613323132928, 280917704704000, 6777200695050240, 163215697915150336, 3926183399462535168, 94372512377130188800

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..2} A075513(3,m)*exp(8*(m+1)*x)/2!.

Cf. A060195, A076004.

a(n) = A075503(n+3, 3) = (8^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (8^n - 8*16^n + 9*24^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 8*k*x).
E.g.f.: (d^3/dx^3)(((exp(8*x)-1)/8)^3)/3! = (exp(8*x) - 8*exp(16*x) + 9*exp(24*x))/2!.

cp's OEIS Frontend

A016140 Expansion of 1/((1-3*x)*(1-8*x)).

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A075503 Stirling2 triangle with scaled diagonals (powers of 8).

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A065128 Number of invertible n X n matrices mod 4 (i.e., over the ring Z_4).

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A076003 Third column of triangle A075503.

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A016140 Expansion of 1/((1-3x)(1-8*x)).