A027876 -id:A027876 - cp's OEIS frontend

A027871 a(n) = Product_{i=1..n} (3^i - 1).

1, 2, 16, 416, 33280, 8053760, 5863137280, 12816818094080, 84078326697164800, 1654829626053597593600, 97714379759212830706892800, 17309711516825516108403231948800

Offset: 0

N. J. A. Sloane

nonn

2*(10)^m|a(n) where 4*m <= n <= 4*m+3 for m >= 1. - G. C. Greubel, Nov 20 2015

Given probability p = 1/3^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1-a(n)/A047656(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100220 ~ 0.4398739. - Bob Selcoe, Mar 01 2016

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

Cf. A005329 (q=2), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A047656, A100220, A132324.

[1] cat [&*[ 3^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027871 := proc(n)
    mul( 3^i-1,i=1..n) ;
end proc:
seq(A027871(n),n=0..8) ; # R. J. Mathar, Jul 13 2017

Table[Product[(3^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[3, 3, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = prod(i=1, n, 3^i-1); \\ Michel Marcus, Nov 21 2015

a(n) ~ c * 3^(n*(n+1)/2), where c = A100220 = Product_{k>=1} (1-1/3^k) = 0.560126077927948944969792243314140014379736333798... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 3^(binomial(n+1,2))*(1/3;1/3){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024023(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 3^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 3^k*x). - Ilya Gutkovskiy, May 22 2017
From Amiram Eldar, Feb 19 2022: (Start)
Sum_{n>=0} 1/a(n) = A132324.
Sum_{n>=0} (-1)^n/a(n) = A100220. (End)

A027637 a(n) = Product_{i=1..n} (4^i - 1).

Original entry on oeis.org

1, 3, 45, 2835, 722925, 739552275, 3028466566125, 49615367752825875, 3251543125681443718125, 852369269595510700600441875, 893773106866112632882108339078125, 3748755223447856814435325652920396921875

Offset: 0

N. J. A. Sloane

The q-analog of double factorials (A000165) evaluated at q=2. - Michael Somos, Sep 12 2014
3^n*5^(floor(n/2))|a(n) for n>=1. - G. C. Greubel, Nov 21 2015
Given probability p = 1/4^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1-a(n)/A053763(n+1) is the probability that the outcome has occurred up to and including the n-th iteration. The limiting ratio is 1-A100221 ~ 0.3114625. - Bob Selcoe, Mar 01 2016

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

Cf. A000165.
Sequences of the form q-Pochhammer(n, q, q): A005329 (q=2), A027871 (q=3), this sequence (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).
Cf. A053763, A100221.

[1] cat [&*[4^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027637 := proc(n)
    mul( 4^i-1,i=1..n) ;
end proc:
seq(A027637(n),n=0..8) ;

A027637 = Table[Product[4^i - 1, {i, n}], {n, 0, 9}] (* Alonso del Arte, Nov 14 2011 *)
a[ n_] := If[ n < 0, 0, Product[ (q^(2 k) - 1) / (q - 1), {k, n}] /. q -> 2]; (* Michael Somos, Sep 12 2014 *)
Abs@QPochhammer[4, 4, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = prod(i=1, n, 4^i-1); \\ Michel Marcus, Nov 21 2015

from sage.combinat.q_analogues import q_pochhammer
def A027637(n): return (-1)^n*q_pochhammer(n, 4, 4)
[A027637(n) for n in (0..15)] # G. C. Greubel, Aug 04 2022

a(n) ~ c * 2^(n*(n+1)), where c = Product_{k>=1} (1-1/4^k) = A100221 = 0.688537537120339715456514357293508184675549819378... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 4^(binomial(n+1,2))*(1/4;1/4){n} = (4; 4){n}, where (a;q){n} is the q-Pochhammer symbol. - _G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 4^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 4^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A100221. - Amiram Eldar, May 07 2023

A027878 a(n) = Product_{i=1..n} (10^i - 1).

Original entry on oeis.org

1, 9, 891, 890109, 8900199891, 890011088900109, 890010198889020099891, 8900101098880002109889900109, 890010100987899112108987901010099891, 890010100097889011121088788901111989989900109

Offset: 0

N. J. A. Sloane

nonn

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), A027879 (q=11), A027880 (q=12).

Cf. A132038, A132326.

[1] cat [&*[10^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Table[Product[10^i-1,{i,n}],{n,0,10}] (* Harvey P. Dale, Aug 15 2011 *)
Abs@QPochhammer[10, 10, Range[0, 30]] (* G. C. Greubel, Nov 24 2015 *)

a(n) = prod(k=1, n, 10^k - 1) \\ Altug Alkan, Nov 25 2015

a(n) ~ c * 10^(n*(n+1)/2), where c = Product_{k>=1} (1-1/10^k) = A132038 = 0.890010099998999000000100009999999989999900000000... . - Vaclav Kotesovec, Nov 21 2015
3^n*(11)^(floor(n/2)) divides a(n) for n>=0. - G. C. Greubel, Nov 24 2015
Equals 10^(binomial(n+1,2))*(1/10;1/10){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 10^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 10^k*x). - Ilya Gutkovskiy, May 22 2017
From Amiram Eldar, May 07 2023: (Start)
Sum_{n>=0} 1/a(n) = A132326.
Sum_{n>=0} (-1)^n/a(n) = A132038. (End)

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

Original entry on oeis.org

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

A027873 a(n) = Product_{i=1..n} (6^i - 1).

Original entry on oeis.org

1, 5, 175, 37625, 48724375, 378832015625, 17674407688984375, 4947685316415841015625, 8310206472731792807458984375, 83747726219216824716765369541015625

Offset: 0

N. J. A. Sloane

nonn

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), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A132034.

[1] cat [&*[ 6^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Table[Product[(6^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[6, 6, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
FoldList[Times,Join[{1},6^Range[10]-1]] (* Harvey P. Dale, Oct 13 2017 *)

a(n) = prod(i=1, n, 6^i-1); \\ Michel Marcus, Nov 21 2015

5^n|a(n) for n>=0. - G. C. Greubel, Nov 20 2015
a(n) ~ c * 6^(n*(n+1)/2), where c = Product_{k>=1} (1-1/6^k) = A132034 = 0.805687728162164940923750215496298968917997628693... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 6^(binomial(n+1,2))*(1/6;1/6){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024062(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 6^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 6^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132034. - Amiram Eldar, May 07 2023

A027875 a(n) = Product_{i=1..n} (7^i - 1).

Original entry on oeis.org

1, 6, 288, 98496, 236390400, 3972777062400, 467389275837235200, 384914699001548351078400, 2218956256804125934296760320000, 89542886518308517126993353029713920000

Offset: 0

N. J. A. Sloane

nonn

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), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A132035.

[1] cat [&*[ 7^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Abs@QPochhammer[7, 7, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
Table[Product[7^k-1,{k,n}],{n,0,10}] (* Harvey P. Dale, Jul 28 2022 *)

a(n) = prod(i=1, n, 7^i-1); \\ Michel Marcus, Nov 21 2015

2*(10)^(2m)|a(n) where 4*m <= n <= 4*m+3, for m >= 1. - G. C. Greubel, Nov 20 2015
a(n) ~ c * 7^(n*(n+1)/2), where c = Product_{k>=1} (1-1/7^k) = A132035 = 0.836795407089037871026729798146136241352436435876... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 7^(binomial(n+1,2))*(1/7;1/7){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024075(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 7^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 7^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132035. - Amiram Eldar, May 07 2023

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A027871 a(n) = Product_{i=1..n} (3^i - 1).

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A027637 a(n) = Product_{i=1..n} (4^i - 1).

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A027878 a(n) = Product_{i=1..n} (10^i - 1).

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

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PARI

Formula

A027873 a(n) = Product_{i=1..n} (6^i - 1).

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Magma

Mathematica

PARI

Formula

A027875 a(n) = Product_{i=1..n} (7^i - 1).

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Author

Keywords

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Programs

Magma

Mathematica

PARI

Formula

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

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