A027873 -id:A027873 - cp's OEIS frontend

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

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

A022170 Triangle of Gaussian binomial coefficients [ n,k ] for q = 6.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 43, 43, 1, 1, 259, 1591, 259, 1, 1, 1555, 57535, 57535, 1555, 1, 1, 9331, 2072815, 12485095, 2072815, 9331, 1, 1, 55987, 74630671, 2698853335, 2698853335, 74630671, 55987, 1, 1, 335923

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  1,     1;
  1,     7,        1;
  1,    43,       43,          1;
  1,   259,     1591,        259,          1;
  1,  1555,    57535,      57535,       1555,        1;
  1,  9331,  2072815,   12485095,    2072815,     9331,     1;
  1, 55987, 74630671, 2698853335, 2698853335, 74630671, 55987, 1 ;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.

G. C. Greubel, Rows n=0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Index entries for sequences related to Gaussian binomial coefficients

Cf. A003463 (k=1), A022220 (k=2), A022221 (k=3).

A027873 := proc(n)
    mul(6^i-1,i=1..n) ;
end procc:
A022170 := proc(n,m)
    A027873(n)/A027873(m)/A027873(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

p[n_]:= Product[6^i - 1, {i, 1, n}]; t[n_, k_]:= p[n]/(p[k]*p[n-k]); Table[t[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Aug 13 2016 *)
Table[QBinomial[n,k,6], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 6; T[n_, 0]:= 1; T[n_,n_]:= 1; T[n_,k_]:= T[n,k] = If[k < 0 || n < k, 0, T[n-1, k -1] +q^k*T[n-1,k]]; Table[T[n,k], {n,0,10}, {k,0,n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

{q=6; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017

G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 6^j - 1. - Seiichi Manyama, May 09 2025

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

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

A220790 Product(6^n - 6^k, k=0..n-1).

Original entry on oeis.org

1, 5, 1050, 8127000, 2273284440000, 22906523331216000000, 8310241106635054164480000000, 108537128570336598656772717772800000000, 51032497739317419104816901041614046625792000000000

Offset: 0

Vincenzo Librandi, Jan 28 2013

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

Cf. A027873, A109354.

Sequences given by product(m^n-m^k, k=0..n-1): A002884 (m=2), A053290 (m=3), A053291 (m=4), A053292 (m=5), A053293 (m=7), A052496 (m=8), A052497 (m=9), A052498 (m=11).

[1] cat [&*[(6^n - 6^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

/* By the second formula: */
m:=9;
A109354 := [6^(n*(n-1) div 2): n in [0..m-1]];
A027873 := [1] cat [&*[6^i-1: i in [1..n]]: n in [1..m]];
[A109354[i]*A027873[i]: i in [1..m]]; // Bruno Berselli, Jan 30 2013

Table[Product[6^n - 6^k, {k, 0, n-1}], {n, 0, 60}]

a(n) = (6^n - 1)*(6^n - 6)*...*(6^n - 6^(n-1)) for n>0, a(0)=1.

a(n) = A109354(n)*A027873(n). - Bruno Berselli, Jan 30 2013

A320354 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = Product_{j=1..n} (k^j - 1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 16, 21, 0, 1, 4, 45, 416, 315, 0, 1, 5, 96, 2835, 33280, 9765, 0, 1, 6, 175, 11904, 722925, 8053760, 615195, 0, 1, 7, 288, 37625, 7428096, 739552275, 5863137280, 78129765, 0, 1, 8, 441, 98496, 48724375, 23205371904, 3028466566125, 12816818094080, 19923090075, 0

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

			Square array begins:
  1,     1,        1,          1,            1,             1,  ...
  0,     1,        2,          3,            4,             5,  ...
  0,     3,       16,         45,           96,           175,  ...
  0,    21,      416,       2835,        11904,         37625,  ...
  0,   315,    33280,     722925,      7428096,      48724375,  ...
  0,  9765,  8053760,  739552275,  23205371904,  378832015625,  ...

Columns k=1..12 give A000007, A005329, A027871, A027637, A027872, A027873, A027875, A027876, A027877, A027878, A027879, A027880.

Cf. A069777, A225816.

Table[Function[k, Product[k^j - 1, {j, 1, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[k^(i (i + 1)/2) x^i/Product[(1 + k^j x), {j, 0, i}], {i, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 9}, {n, 0, m}] // Flatten

G.f. of column k: Sum_{i>=0} k^(i*(i+1)/2)*x^i / Product_{j=0..i} (1 + k^j*x).

For asymptotics of column k see comment from Vaclav Kotesovec in A027880.

cp's OEIS Frontend

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

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A022170 Triangle of Gaussian binomial coefficients [ n,k ] for q = 6.

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

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