A027637 -id:A027637 - cp's OEIS frontend

A027879 a(n) = Product_{i=1..n} (11^i - 1).

1, 10, 1200, 1596000, 23365440000, 3763004112000000, 6666387564654720000000, 129909027758312519942400000000, 27847153692160782464830528512000000000, 65662131721505488121539650946349537280000000000

Offset: 0

N. J. A. Sloane

nonn

It appears that the number of trailing zeros in a(n) is A191610(n). - Robert Israel, Nov 24 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..43

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027880 (q=12).

Cf. A132267, A191610.

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

seq(mul(11^i-1,i=1..n),n=0..20; # Robert Israel, Nov 24 2015

FoldList[Times,1,11^Range[10]-1] (* Harvey P. Dale, Aug 13 2013 *)
Abs@QPochhammer[11, 11, Range[0, 40]] (* G. C. Greubel, Nov 24 2015 *)

a(n)=prod(i=1,n,11^i-1) \\ Anders Hellström, Nov 21 2015

10^n|a(n) for n>=0; 12*(10)^(n)|a(n) n>=2. - G. C. Greubel, Nov 21 2015
a(n) ~ c * 11^(n*(n+1)/2), where c = Product_{k>=1} (1-1/11^k) = 0.900832706809715279949862694760647744762491192216... . - Vaclav Kotesovec, Nov 21 2015
E.g.f. E(x) satisfies E'(x) = 11 E(11 x) - E(x). - Robert Israel, Nov 24 2015
Equals 11^(binomial(n+1,2))*(1/11;1/11){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 11^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 11^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132267. - Amiram Eldar, May 07 2023

A027880 a(n) = Product_{i=1..n} (12^i - 1).

Original entry on oeis.org

1, 11, 1573, 2716571, 56328099685, 14016177372718235, 41852067359921313500005, 1499635200191700040518673659035, 644815685260091508353787979063721364325, 3327107302821620489265827570792988872583047378075

Offset: 0

N. J. A. Sloane

nonn

In general, Product_{i=1..n} (q^i-1) ~ c * q^(n*(n+1)/2), where c = Product_{k >= 1} (1-1/q^k). - Vaclav Kotesovec, Nov 21 2015

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

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11).

Cf. A132268.

[1] cat [&*[12^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

FoldList[Times,1,12^Range[10]-1] (* Harvey P. Dale, Mar 01 2015 *)
Abs@QPochhammer[12, 12, Range[0, 30]] (* G. C. Greubel, Nov 24 2015 *)

a(n) = prod(k=1, n, 12^k - 1) \\ Altug Alkan, Nov 25 2015

a(n) ~ c * 12^(n*(n+1)/2), where c = Product_{k>=1} (1-1/12^k) = 0.909726268905994888636362046977080249120791691941... . - Vaclav Kotesovec, Nov 21 2015
(11)^n*(13)^(floor(n/2))|a(n) for n>=0. - G. C. Greubel, Nov 24 2015
Equals 12^(binomial(n+1,2))*(1/12;1/12){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 12^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 12^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132268. - Amiram Eldar, May 07 2023

A053291 Nonsingular n X n matrices over GF(4).

Original entry on oeis.org

1, 3, 180, 181440, 2961100800, 775476766310400, 3251791214634074112000, 218210695042457748180566016000, 234298374547168764346587444978647040000, 4025200069765920285793155323595159699896401920000, 1106437515026051855463365435310419366987397763763798016000000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..40

Cf. A002884, A003788, A027637, A053763, A053290, A053292, A053293.

Cf. A100221, A115490.

[1] cat [&*[(4^n - 4^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

Table[Product[4^n - 4^k, {k,0,n-1}], {n,0,10}] (* Geoffrey Critzer, Jan 26 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 4^n - 4^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (4^n - 1)*(4^n - 4)*...*(4^n - 4^(n-1)).
a(n) = A053763(n)*A027637(n). - Bruno Berselli, Jan 30 2013
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} A115490(k).
a(n) ~ c * 4^(n^2), where c = A100221. (End)

More terms from Vladeta Jovovic, Mar 16 2000

A028692 23-factorial numbers.

Original entry on oeis.org

1, 22, 11616, 141320256, 39547060439040, 254538406080331591680, 37680818974206486508802211840, 128296611269497862923425473853914480640, 10047034036599529256387830050150921763777884979200, 18096242094820543236399273859296273669601076798103392511590400

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..38
Index entries for sequences related to factorial numbers.

Cf. A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880, A028693, A028694.

FoldList[ #1 (23^#2-1)&, 1, Range[ 20 ] ]
a[n_] := Abs[QPochhammer[23, 23, n]]; Array[a, 10, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = prod(k = 1, n, 23^k - 1); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = Product_{k=1..n} (23^k-1).
a(n) ~ c * 23^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/23^k) = 0.954631535623... . (End)

A028693 24-factorial numbers.

Original entry on oeis.org

1, 23, 13225, 182809175, 60651514035625, 482945140644890444375, 92292253139031982469134515625, 423295781586452233477722435457009484375, 46594416147080909523690749946376478698532878515625, 123093479909646650570543074660375014342475500150254964721484375

Offset: 1

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..38
Index entries for sequences related to factorial numbers.

Cf. A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880, A028692, A028694.

FoldList[ #1 (24^#2-1)&, 1, Range[ 20 ] ]
a[n_] := Abs[QPochhammer[24, 24, n]]; Array[a, 10, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = prod(k = 1, n, 24^k - 1); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = Product_{k=1..n} (24^k-1).
a(n) ~ c * 24^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/24^k) = 0.956597348026... . (End)

A028694 25-factorial numbers.

Original entry on oeis.org

1, 24, 14976, 233985024, 91400166014976, 892579654839833985024, 217914953902301689160166014976, 1330047325845938129350664710839833985024, 202949115880923695556030391039325175289160166014976, 774189437411767935420978172981557217629743778824710839833985024

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..37
Index entries for sequences related to factorial numbers.

Cf. A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880, A028692, A028693.

FoldList[ #1 (25^#2-1)&, 1, Range[ 20 ] ]
a[n_] := Abs[QPochhammer[25, 25, n]]; Array[a, 10, 0] (* Amiram Eldar, Jul 14 2025 *)

a(n) = prod(k = 1, n, 25^k - 1); \\ Amiram Eldar, Jul 14 2025

From Amiram Eldar, Jul 14 2025: (Start)
a(n) = Product_{k=1..n} (25^k-1).
a(n) ~ c * 25^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/25^k) = 0.958400102563... . (End)

A263394 a(n) = Product_{i=1..n} (3^i - 2^i).

Original entry on oeis.org

1, 5, 95, 6175, 1302925, 866445125, 1784010512375, 11248186280524375, 215638979183932793125, 12512451767147700321078125, 2190917791975795178520458609375, 1155369543009475708416871245360859375, 1832567448623162714866960405275465241328125

Offset: 1

Bob Selcoe, Mar 02 2016

Generally, for sequences of the form a(n) = Product_{i=1..n} j^i-k^i, where j>k>=1 and n>=1: given probability p=(k/j)^n that an outcome will occur at the n-th stage of an infinite process, then r = 1 - a(n)/j^((n^2+n)/2) is the probability that the outcome has occurred up to and including the n-th iteration. Here, j=3 and k=2, so p=(2/3)^n and r = 1-a(n)/A047656(n+1). The limiting ratio of r ~ 0.9307279.

Cf. A001047, A047656.
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: A269576 (j=4, k=3), A269661 (j=5, k=4).

[&*[ 3^k-2^k: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Mar 03 2016

A263394:=n->mul(3^i-2^i, i=1..n): seq(A263394(n), n=1..15); # Wesley Ivan Hurt, Mar 02 2016

Table[Product[3^i - 2^i, {i, n}], {n, 15}] (* Wesley Ivan Hurt, Mar 02 2016 *)
FoldList[Times,Table[3^i-2^i,{i,15}]] (* Harvey P. Dale, Feb 06 2017 *)

a(n) = prod(k=1, n, 3^k-2^k); \\ Michel Marcus, Mar 05 2016

a(n) = Product_{i=1..n} A001047(i).

a(n) ~ c * 3^(n*(n+1)/2), where c = QPochhammer(2/3) = 0.0692720728018644... . - Vaclav Kotesovec, Oct 10 2016

A269576 a(n) = Product_{i=1..n} (4^i - 3^i).

Original entry on oeis.org

1, 7, 259, 45325, 35398825, 119187843775, 1692109818073675, 99792176520894983125, 24195710911432718503470625, 23942309231057283642583777144375, 96180015123706384385790918441966041875

Offset: 1

Bob Selcoe, Mar 02 2016

nonn

In general, for sequences of the form a(n) = Product_{i=1..n} j^i-k^i, where j>k>=1 and n>=1: given probability p=(k/j)^n that an outcome will occur at the n-th stage of an infinite process, then r = 1 - a(n)/j^((n^2+n)/2) is the probability that the outcome has occurred at or before the n-th iteration. Here j=4 and k=3, so p=(3/4)^n and r = 1-a(n)/A053763(n+1). The limiting ratio of r is ~ 0.9844550.

Robert Israel, Table of n, a(n) for n = 1..57

Cf. A005061, A053763.
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), A269661 (j=5, k=4).

seq(mul(4^i-3^i,i=1..n),n=0..20); # Robert Israel, Jun 01 2023

Table[Product[4^i - 3^i, {i, n}], {n, 11}] (* Michael De Vlieger, Mar 07 2016 *)
FoldList[Times,Table[4^n-3^n,{n,20}]] (* Harvey P. Dale, Jul 30 2018 *)

a(n) = prod(k=1, n, 4^k-3^k); \\ Michel Marcus, Mar 05 2016

a(n) = Product_{i=1..n} A005061(i).
a(n) ~ c * 2^(n*(n+1)), where c = QPochhammer(3/4) = 0.015545038845451847... . - Vaclav Kotesovec, Oct 10 2016
a(n+3)/a(n+2) - 7 * a(n+2)/a(n+1) + 12 * a(n+1)/a(n) = 0. - Robert Israel, Jun 01 2023

A309327 a(n) = Product_{k=1..n-1} (4^k + 1).

Original entry on oeis.org

1, 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, 1678532740564688125136703125, 1760070825503098980191468752703125, 7382273863761775568111978346806480703125, 123854010565759745011512941023673583762640703125

Offset: 0

Ilya Gutkovskiy, Jun 06 2020

nonn

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

Cf. A027637, A052539, A053763.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), A290000 (m=3), this sequence (m=4).

[n lt 2 select 1 else (&*[4^j +1: j in [1..n-1]]): n in [0..15]]; // G. C. Greubel, Feb 21 2021

Table[Product[4^k + 1, {k, 1, n - 1}], {n, 0, 13}]
Join[{1}, Table[4^(Binomial[n,2])*QPochhammer[-1/4, 1/4, n-1], {n,15}]] (* G. C. Greubel, Feb 21 2021 *)

a(n) = prod(k=1, n-1, 4^k + 1); \\ Michel Marcus, Jun 06 2020

from sage.combinat.q_analogues import q_pochhammer
[1]+[4^(binomial(n,2))*q_pochhammer(n-1, -1/4, 1/4) for n in (1..15)] # G. C. Greubel, Feb 21 2021

G.f. A(x) satisfies: A(x) = 1 + x * A(4*x) / (1 - x).
G.f.: Sum_{k>=0} 2^(k*(k - 1)) * x^k / Product_{j=0..k-1} (1 - 4^j*x).
a(0) = 1; a(n) = Sum_{k=0..n-1} 4^k * a(k).
a(n) ~ c * 2^(n*(n - 1)), where c = Product_{k>=1} (1 + 1/4^k) = 1.355909673863479380345544...
a(n) = 4^(binomial(n+1,2))*(-1/4; 1/4){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021

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

cp's OEIS Frontend

A027879 a(n) = Product_{i=1..n} (11^i - 1).

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A027880 a(n) = Product_{i=1..n} (12^i - 1).

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A053291 Nonsingular n X n matrices over GF(4).

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A028692 23-factorial numbers.

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A028693 24-factorial numbers.

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A028694 25-factorial numbers.

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A263394 a(n) = Product_{i=1..n} (3^i - 2^i).

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