A027878 -id:A027878 - cp's OEIS frontend

A027877 a(n) = Product_{i=1..n} (9^i - 1).

1, 8, 640, 465920, 3056435200, 180476385689600, 95912370410881024000, 458745798479390789599232000, 19747501938318761090457052119040000, 7650586837724400321220283274999910891520000

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..40

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A132037.

[1] cat [&*[ 9^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Abs@QPochhammer[9, 9, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = prod(i=1, n, 9^i-1); \\ Altug Alkan, Dec 24 2015

a(n) ~ c * 3^(n*(n+1)), where c = Product_{k>=1} (1-1/9^k) = A132037 = 0.876560354035964205836019838417862010106635101174... . - Vaclav Kotesovec, Nov 21 2015
From  - G. C. Greubel, Dec 24 2015: (Start)
8^n * 10^(floor(n/2))|a(n), for n>=0.
a(n) = 9^(binomial(n+1,2))*(1/9;1/9){n}, where (a;q){n} is the q-Pochhammer symbol. (End)
a(n) = Product_{i=1..n} A024101(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 9^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 9^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132037. - Amiram Eldar, May 07 2023

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

Original entry on oeis.org

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

A022172 Triangle of Gaussian binomial coefficients [ n,k ] for q = 8.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 73, 73, 1, 1, 585, 4745, 585, 1, 1, 4681, 304265, 304265, 4681, 1, 1, 37449, 19477641, 156087945, 19477641, 37449, 1, 1, 299593, 1246606473, 79936505481, 79936505481, 1246606473, 299593, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  1,      1;
  1,      9,          1;
  1,     73,         73,           1;
  1,    585,       4745,         585,           1;
  1,   4681,     304265,      304265,        4681,          1;
  1,  37449,   19477641,   156087945,    19477641,      37449,      1;
  1, 299593, 1246606473, 79936505481, 79936505481, 1246606473, 299593, 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. A023001 (k=1), A022242 (k=2).

A027876 := proc(n)
    mul(8^i-1,i=1..n) ;
end proc:
A022172 := proc(n,m)
    A027876(n)/A027876(m)/A027876(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

a027878[n_]:=Times@@ Table[8^i - 1, {i, n}]; T[n_, m_]:=a027878[n]/( a027878[m] a027878[n - m]); Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* Indranil Ghosh, Jul 20 2017 *)
Table[QBinomial[n,k,8], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 8; 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=8; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

from operator import mul
def a027878(n): return 1 if n==0 else reduce(mul, [8**i - 1 for i in range(1, n + 1)])
def T(n, m): return a027878(n)//(a027878(m)*a027878(n - m))
for n in range(11): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, Jul 20 2017

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) = 8^j - 1. - Seiichi Manyama, May 09 2025

A022174 Triangle of Gaussian binomial coefficients [ n,k ] for q = 10.

Original entry on oeis.org

1, 1, 1, 1, 11, 1, 1, 111, 111, 1, 1, 1111, 11211, 1111, 1, 1, 11111, 1122211, 1122211, 11111, 1, 1, 111111, 112232211, 1123333211, 112232211, 111111, 1, 1, 1111111, 11223332211, 1123445443211, 1123445443211, 11223332211, 1111111, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  1,       1;
  1,      11,           1;
  1,     111,         111,             1;
  1,    1111,       11211,          1111,             1;
  1,   11111,     1122211,       1122211,         11111,           1;
  1,  111111,   112232211,    1123333211,     112232211,      111111,       1;
  1, 1111111, 11223332211, 1123445443211, 1123445443211, 11223332211, 1111111, 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
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

Row sums give A015196.

A027878 := proc(n)
    mul(10^i-1,i=1..n) ;
end proc:
A022174 := proc(n,m)
    A027878(n)/A027878(m)/A027878(n-m) ;
end proc:# R. J. Mathar, Jul 19 2017

a027878[n_]:=Times@@ Table[10^i - 1, {i, n}]; T[n_, m_]:=a027878[n]/( a027878[m] a027878[n - m]); Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* Indranil Ghosh, Jul 20 2017, after Maple code *)
Table[QBinomial[n,k,10], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 10; 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 28 2018 *)

{q=10; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 28 2018

from operator import mul
def a027878(n): return 1 if n==0 else reduce(mul, [10**i - 1 for i in range(1, n + 1)])
def T(n, m): return a027878(n)/(a027878(m)*a027878(n - m))
for n in range(11): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, Jul 20 2017, after Maple code

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) = 10^j - 1. - Seiichi Manyama, May 09 2025

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

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

cp's OEIS Frontend

A027877 a(n) = Product_{i=1..n} (9^i - 1).

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

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A022172 Triangle of Gaussian binomial coefficients [ n,k ] for q = 8.

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A022174 Triangle of Gaussian binomial coefficients [ n,k ] for q = 10.

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

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A028692 23-factorial numbers.

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Mathematica

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A028693 24-factorial numbers.

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