A109345 -id:A109345 - cp's OEIS frontend

A027872 a(n) = Product_{i=1..n} (5^i - 1).

1, 4, 96, 11904, 7428096, 23205371904, 362560730628096, 28324694519589371904, 11064305472020078810628096, 21609960560733744406929189371904, 211034749490954911990173458030810628096

Offset: 0

N. J. A. Sloane

nonn

Given probability p = 1/5^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1 - a(n)/A109345(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100222 ~ 0.2396672. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A109345, A100222.

[1] cat [&*[ 5^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027872 := proc(n)
        mul( 5^i-1, i=1..n) ;
end proc: # R. J. Mathar, Mar 12 2013

Table[Product[(5^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[5, 5, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
Join[{1},FoldList[Times,5^Range[10]-1]] (* Harvey P. Dale, Dec 28 2021 *)

a(n) = prod(i=1, n, 5^i-1); \\ Michel Marcus, Nov 21 2015

4^n|a(n) for n >= 1. - G. C. Greubel, Nov 21 2015
a(n) ~ c * 5^(n*(n+1)/2), where c = Product_{k>=1} (1-1/5^k) = A100222 . - Vaclav Kotesovec, Nov 21 2015
a(n) = 5^(binomial(n+1,2))*(1/5; 1/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
a(n) = Product_{i=1..n} A024049(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 5^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 5^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A100222. - Amiram Eldar, May 07 2023

A053292 Number of nonsingular n X n matrices over GF(5).

Original entry on oeis.org

1, 4, 480, 1488000, 116064000000, 226614960000000000, 11064475422000000000000000, 13506266841692625000000000000000000, 412177498341354683437500000000000000000000000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..35
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=5 of A316622.

Cf. A002884, A003789, A027872, A053290, A053291, A053293, A109345, A100222.

[1] cat [&*[(5^n - 5^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

Table[Product[5^n - 5^k, {k,0,n-1}], {n,0,10}] (* Geoffrey Critzer, Jan 26 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 5^n - 5^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (5^n - 1)*(5^n - 5)*...*(5^n - 5^(n-1)).
a(n) = A109345(n)*A027872(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 5^(n^2), where c = A100222. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A109354 a(n) = 6^((n^2 - n)/2).

Original entry on oeis.org

1, 1, 6, 216, 46656, 60466176, 470184984576, 21936950640377856, 6140942214464815497216, 10314424798490535546171949056, 103945637534048876111514866313854976, 6285195213566005335561053533150026217291776, 2280250319867037997421842330085227917956272625811456

Offset: 0

Philippe Deléham, Aug 25 2005

nonn

Sequence given by the Hankel transform (see A001906 for definition) of A078018 = {1, 1, 7, 55, 469, 4237, 39907, 387739, ...}; example: det([1, 1, 7, 55; 1, 7, 55, 469; 7, 55, 469, 4237; 55, 469, 4237, 39907]) = 6^6 = 46656.

In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 6 types of edge. - Mark Stander, Apr 11 2019

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000400, A006125, A047656, A053763, A053764, A109345.

Table[6^((n^2-n)/2),{n,0,10}] (* Harvey P. Dale, May 28 2013 *)

a(n) = 6^((n^2 - n)/2); \\ Michel Marcus, Apr 12 2019

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(6i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(6*x). - Ilya Gutkovskiy, Jun 04 2020

Terms a(11) and beyond from Andrew Howroyd, Jan 02 2020

A157832 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (5^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 5, -6, 1, 125, -155, 31, -1, 15625, -19500, 4030, -156, 1, 9765625, -12203125, 2538250, -101530, 781, -1, 30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1, 476837158203125, -596038818359375

Offset: 0

Roger L. Bagula, Mar 07 2009

Except for n=0, the row sums are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=5)= [1,4,25,120,625,3100,15625,...] DELTA [ -1,0,-5,0,-25,0,-125,0,-625,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins
  1;
  1, -1;
  5, -6, 1;
  125, -155, 31, -1;
  15625, -19500, 4030, -156, 1;
  9765625, -12203125, 2538250, -101530, 781, -1;
  30517578125, -38144531250, 7944234375, -319819500, 2542155, -3906, 1;
  476837158203125, -596038818359375, 124166806640625, -5005123921875, 40040991375, -63573405, 19531, -1;

Cf. A135950, A157783, A109345 (first column), A003463 (first subdiagonal).

A157832 := proc(n,k)
        product( 5^(i-1)-x,i=1..n) ;
        coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

p[x_, n_] = If[n == 0, 1, Product[q^(i - 1) - x, {i, 1, n}]];
q = 5;
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

A109493 a(n) = 7^((n^2 - n)/2).

Original entry on oeis.org

1, 1, 7, 343, 117649, 282475249, 4747561509943, 558545864083284007, 459986536544739960976801, 2651730845859653471779023381601, 107006904423598033356356300384937784807

Offset: 0

Philippe Deléham, Aug 29 2005

Sequence given by the Hankel transform (see A001906 for definition) of A081178 = {1, 1, 8, 71, 680, 6882, 72528, 788019, ...}; example: det([1, 1, 8, 71; 1, 8, 71, 680; 8, 71, 680, 6882; 71, 680, 6882, 72528]) = 7^6 = 117649.

In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 7 types of edge. - Mark Stander, Apr 11 2019

Cf. A006125, A047656, A053763, A053764, A109345, A109354.

a(n) = 7^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(7i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(7*x). - Ilya Gutkovskiy, Jun 04 2020

A109966 a(n) = 8^((n^2-n)/2).

Original entry on oeis.org

1, 1, 8, 512, 262144, 1073741824, 35184372088832, 9223372036854775808, 19342813113834066795298816, 324518553658426726783156020576256, 43556142965880123323311949751266331066368, 46768052394588893382517914646921056628989841375232, 401734511064747568885490523085290650630550748445698208825344

Offset: 0

Philippe Deléham, Sep 01 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082147 = {1, 1, 9, 89, 945, 10577, 123129, 1476841, ...}; example: det([1, 1, 9, 89; 1, 9, 89, 945; 9, 89, 945, 10577; 89, 945, 10577, 123129]) = 8^6 = 262144.

The number of labeled multigraphs on n vertices such that (i) no self loops are allowed; (ii) all edges are painted in one of 3 colors; (iii) edges between any pair of vertices are painted in distinct colors. Note, this implies that there are at most 3 edges between any vertex pair. Also note there is no restriction on the color of edges incident to a common vertex. - Geoffrey Critzer, Jan 14 2020

Cf. A006125, A047656, A053763, A053764, A059435, A082147, A109345, A109354, A109493.

[2^(3*Binomial(n,2)): n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[2^(3*Binomial[n,2]),{n,0,10}] (* Geoffrey Critzer, Nov 10 2011 *)

a(n)=8^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(8i, j).

Hankel transform of A059435. - Philippe Deléham, Sep 03 2006

a(10) corrected and a(11), a(12) from Georg Fischer, Apr 01 2022

A110147 10^((n^2-n)/2).

Original entry on oeis.org

1, 1, 10, 1000, 1000000, 10000000000, 1000000000000000, 1000000000000000000000, 10000000000000000000000000000, 1000000000000000000000000000000000000

Offset: 0

Philippe Deléham, Sep 04 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082148 = {1, 1, 11, 131, 1661, 22101, 305151, 4335711, ...}; example: det([1, 1, 11, 131; 1, 11, 131, 1661; 11, 131, 1661, 22101; 131, 1661, 22101, 305151]) = 10^6 = 1000000.

Also the Hankel transform of A379103. - Nathaniel Johnston, Dec 16 2024

Cf. A006125, A047656, A053763, A053764, A109345, A109354, A109493, A109966.

a(n)=10^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(10i, j).

a(n)=10a(n-1)^2/a(n-2), a(0)=a(1)=1. - Michael Somos, Sep 12 2005

A269661 a(n) = Product_{i=1..n} (5^i - 4^i).

Original entry on oeis.org

1, 9, 549, 202581, 425622681, 4907003889249, 302963327126122509, 98490045052104040328301, 166544794872251942218390753281, 1451779137596368920662880897497387769, 64798450159010700654830227323217753649135349

Offset: 1

Bob Selcoe, Mar 02 2016

Cf. A005060, A109345.
Cf. sequences of the form Product_{i=1..n}(j^i - 1): A005329 (j=2), A027871 (j=3), A027637 (j=4), A027872 (j=5), A027873 (j=6), A027875 (j=7), A027876 (j=8), A027877 (j=9), A027878 (j=10), A027879 (j=11), A027880 (j=12).
Cf. sequences of the form Product_{i=1..n}(j^i - k^1), k>1: A263394 (j=3, k=2), A269576 (j=4, k=3).

[&*[ 5^k-4^k: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Mar 03 2016

Table[Product[5^i - 4^i, {i, n}], {n, 15}] (* Vincenzo Librandi, Mar 03 2016 *)
Table[5^(Binomial[n + 1, 2]) *QPochhammer[4/5, 4/5, n], {n, 1, 20}] (* G. C. Greubel, Mar 05 2016 *)
FoldList[Times,Table[5^n-4^n,{n,15}]] (* Harvey P. Dale, Aug 28 2018 *)

a(n) = prod(k=1, n, 5^k-4^k); \\ Michel Marcus, Mar 05 2016

a(n) = Product_{i=1..n} A005060(i).
a(n) = 5^(binomial(n+1,2))*(4/5;4/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Mar 05 2016
a(n) ~ c * 5^(n*(n+1)/2), where c = QPochhammer(4/5) = 0.00336800585242312126... . - Vaclav Kotesovec, Oct 10 2016

A110195 a(n) = 11^((n^2-n)/2).

Original entry on oeis.org

1, 1, 11, 1331, 1771561, 25937424601, 4177248169415651, 7400249944258160101211, 144209936106499234037676064081, 30912680532870672635673352936887453361, 72890483685103052142902866787761839379440139451, 1890591424712781041871514584574319778449301246603238034051

Offset: 0

Philippe Deléham, Sep 07 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082173 = {1, 1, 12, 155, 2124, 30482, 453432, 6936799, ...}; example : det([1, 1, 12, 155; 1, 12, 155, 2124; 12, 155, 2124, 30482; 155, 2124, 30482, 453432]) = 11^6 = 1771561.

Cf. A001020, A006125, A047656, A053763, A053764, A109345, A109354, A109493, A109966, A110147, A161680.

Table[11^((n^2-n)/2),{n,0,20}] (* Harvey P. Dale, Feb 02 2012 *)
Join[{1,1},Table[Det[Table[Binomial[11i,j],{i,n},{j,n}]],{n,10}]] (* Harvey P. Dale, Apr 01 2019 *)

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(11i, j).

a(n) = A001020(A161680(n)).

a(11) from Harvey P. Dale, Feb 02 2012

a(12) from Jason Yuen, Aug 29 2025

cp's OEIS Frontend

A027872 a(n) = Product_{i=1..n} (5^i - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A053292 Number of nonsingular n X n matrices over GF(5).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A109354 a(n) = 6^((n^2 - n)/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A157832 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (5^(i-1)-x) in row n, column k, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A109493 a(n) = 7^((n^2 - n)/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A109966 a(n) = 8^((n^2-n)/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A110147 10^((n^2-n)/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A269661 a(n) = Product_{i=1..n} (5^i - 4^i).

Views