A027878 -id:A027878 - cp's OEIS frontend

A275106 Limiting sequence of the least significant digits of the even-indexed terms of A027878 in reverse order.

1, 9, 8, 9, 9, 0, 0, 1, 0, 0, 0, 0, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9

Offset: 1

Kerry Mitchell, Jul 16 2016

nonn

In other words, take a term A027878(2m) where m is very large, and read the digits from right to left.

Except for n=1, it appears that a(n) + A275107(n) = 9.

Kerry Mitchell, Table of n, a(n) for n = 1..300

A027878 gives the full terms, A275107 gives the limiting sequence of the least significant digits of the odd-indexed terms of A027878 in reverse order.

A275107 Limiting sequence of the least significant digits of the odd-indexed terms of A027878 in reverse order.

Original entry on oeis.org

9, 0, 1, 0, 0, 9, 9, 8, 9, 9, 9, 9, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Kerry Mitchell, Jul 16 2016

nonn

Except for n=1, it appears that A275106(n) + a(n) = 9.

Kerry Mitchell, Table of n, a(n) for n = 1..300

A027878 gives the full terms, A275106 gives the limiting sequence of the least significant digits of the even-indexed terms of A027878 in reverse order.

A132038 Decimal expansion of Product_{k>0} (1-1/10^k).

Original entry on oeis.org

8, 9, 0, 0, 1, 0, 0, 9, 9, 9, 9, 8, 9, 9, 9, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 9

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.8900100999989990000001000...

G. C. Greubel, Table of n, a(n) for n = 0..1500
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, page 49.
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A000203, A027878, A048651, A067080, A098844, A100220, A132019, A132026.

digits = 105; Clear[p]; p[n_] := p[n] = RealDigits[Product[1-1/10^k , {k, 1, n}], 10, digits] // First; p[10]; p[n=20]; While[p[n] != p[n/2], n = 2*n]; p[n] (* Jean-François Alcover, Feb 17 2014 *)
RealDigits[QPochhammer[1/10], 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)
N[QPochhammer[1/10,1/10]] (* G. C. Greubel, Nov 30 2015 *)

prodinf(x=1,-.1^x,1) \\ Charles R Greathouse IV, Nov 16 2013

Equals exp( -Sum_{n>0} sigma_1(n)/(n*10^n) ).
Equals (1/10; 1/10){infinity}, where (a; q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(10)) * exp(log(10)/24 - Pi^2/(6*log(10))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(10))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027878(n). (End)

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

Original entry on oeis.org

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

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

A022173 Triangle of Gaussian binomial coefficients [ n,k ] for q = 9.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 91, 91, 1, 1, 820, 7462, 820, 1, 1, 7381, 605242, 605242, 7381, 1, 1, 66430, 49031983, 441826660, 49031983, 66430, 1, 1, 597871, 3971657053, 322140667123, 322140667123, 3971657053, 597871, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  1,      1;
  1,     10,          1;
  1,     91,         91,            1;
  1,    820,       7462,          820,            1;
  1,   7381,     605242,       605242,         7381,          1;
  1,  66430,   49031983,    441826660,     49031983,      66430,      1;
  1, 597871, 3971657053, 322140667123, 322140667123, 3971657053, 597871, 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

A027877 := proc(n)
    mul(9^i-1,i=1..n) ;
end proc:
A022173 := proc(n,m)
    A027877(n)/A027877(m)/A027877(n-m) ;
end proc: # R. J. Mathar, Jul 19 2017

a027878[n_]:=Times@@ Table[9^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,9], {n,0,10}, {k,0,n}]//Flatten (* or *) q:= 9; 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=9; 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, [9**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) = 9^j - 1. - Seiichi Manyama, May 09 2025

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

A027876 a(n) = Product_{i=1..n} (8^i - 1).

Original entry on oeis.org

1, 7, 441, 225351, 922812345, 30237792108615, 7926625536728661945, 16623330670976050126618695, 278893192683059452825059069034425, 37432410397693271164043156886536608251975

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

Cf. A132036.

[1] cat [&*[ 8^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

seq(mul(8^i-1,i=1..n), n=0..20); # Robert Israel, Dec 24 2015

FoldList[Times,1,8^Range[10]-1] (* Harvey P. Dale, Dec 23 2011 *)

a(n)=prod(i=1,n,8^i-1) \\ Charles R Greathouse IV, Nov 22 2015

a(n) ~ c * 8^(n*(n+1)/2), where c = Product_{k>=1} (1-1/8^k) = A132036 = 0.859405994400702866200758580064418894909484979588... . - Vaclav Kotesovec, Nov 21 2015
7^n | a(n). - G. C. Greubel, Nov 21 2015
It appears that 7^m | a(n) iff 7^m | (7n)!. - Robert Israel, Dec 24 2015
a(n) = 8^(binomial(n+1,2))*(1/8;1/8){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f. g(x) satisfies (1+x) g(x) = 1 + 8 x g(8x). - Robert Israel, Dec 24 2015
a(n) = Product_{i=1..n} A024088(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 8^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 8^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132036. - Amiram Eldar, May 07 2023

cp's OEIS Frontend

A275106 Limiting sequence of the least significant digits of the even-indexed terms of A027878 in reverse order.

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A275107 Limiting sequence of the least significant digits of the odd-indexed terms of A027878 in reverse order.

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A132038 Decimal expansion of Product_{k>0} (1-1/10^k).

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

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A022173 Triangle of Gaussian binomial coefficients [ n,k ] for q = 9.

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