A005329 -id:A005329 - cp's OEIS frontend

A243950 Sum of the squares of q-binomial coefficients for q=2 in row n of triangle A022166, for n >= 0.

1, 2, 11, 100, 1677, 49974, 2801567, 293257480, 59426801521, 23154622451162, 17786849024835651, 26694462878992491180, 79786045619298591331605, 469805503062346255040726910, 5538428985758278544518994721255, 129179377104085570277109465712798800, 6048537751321912538368011648067930447545

Offset: 0

Paul D. Hanna, Jun 21 2014

nonn

a(n) is the number of Green's H classes in the semigroup of n X n matrices over GF(2) (cf. A359313). - Geoffrey Critzer, Jun 20 2023

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + ...
Related integer series:
A(x)^(1/2) = 1 + x + 5*x^2 + 45*x^3 + 781*x^4 + 23981*x^5 + 1371885*x^6 + 145101805*x^7 + 29560055405*x^8 + ... + A243951(n)*x^n + ...
A022166, the triangle of q-binomial coefficients for q=2, begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,     1;
  1,  15,   35,    15,     1;
  1,  31,  155,   155,    31,    1;
  1,  63,  651,  1395,   651,   63,   1;
  1, 127, 2667, 11811, 11811, 2667, 127, 1; ...
from which we can illustrate the initial terms of this sequence:
  a(0) = 1^2 = 1;
  a(1) = 1^2 + 1^2 = 2;
  a(2) = 1^2 + 3^2 + 1^2 = 11;
  a(3) = 1^2 + 7^2 + 7^2 + 1^2 = 100;
  a(4) = 1^2 + 15^2 + 35^2 + 15^2 + 1^2 = 1677;
  a(5) = 1^2 + 31^2 + 155^2 + 155^2 + 31^2 + 1^2 = 49974;
  a(6) = 1^2 + 63^2 + 651^2 + 1395^2 + 651^2 + 63^2 + 1^2 = 2801567; ...

Paul D. Hanna, Table of n, a(n) for n = 0..60
Wikipedia, Green's relations

Cf. A022166, A243951, A359313.

a[n_] := Sum[QBinomial[n, k, 2]^2, {k, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)

{A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
{a(n)=sum(k=0,n,A022166(n, k)^2)}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * 2^(n^2/2), where c = 18.0796893855819714431... if n is even and c = 18.02126069886312898683... if n is odd. - Vaclav Kotesovec, Jun 23 2014

Sum_{n>=0} a(n)*x^n/A005329(n)^2 = E(x)^2 where E(x) = Sum_{n>=0} x^n/A005329(n)^2. - Geoffrey Critzer, Jun 20 2023

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

A335011 Decimal expansion of Product_{k>=1} (1 - 1/2^k)^k.

Original entry on oeis.org

0, 9, 9, 6, 7, 9, 7, 3, 1, 2, 6, 2, 8, 7, 9, 9, 8, 5, 4, 6, 4, 6, 6, 5, 3, 5, 6, 0, 4, 6, 8, 4, 7, 3, 0, 4, 9, 6, 3, 4, 4, 4, 6, 1, 5, 0, 9, 7, 3, 1, 7, 5, 3, 0, 1, 2, 6, 4, 0, 9, 1, 8, 5, 5, 5, 4, 5, 6, 0, 6, 3, 0, 5, 0, 0, 9, 8, 0, 8, 2, 4, 2, 5, 3, 8, 6, 5, 2, 4, 0, 7, 2, 7, 6, 9, 2, 2, 4, 7, 3, 3, 5, 4, 3, 9, 4

Offset: 0

Vaclav Kotesovec, May 19 2020

			0.0996797312628799854646653560468473049634446150973175301264091855545606305...

Cf. A005329, A048651, A203305, A333938, A335010.

evalf(exp(-Sum(2^j / (j * (2^j - 1)^2), j=1..infinity)), 120);

PARI
```
prodinf(k=1, (1 - 1/2^k)^k)
```

Equals exp(-Sum_{j>=1} 2^j / (j * (2^j - 1)^2)).

A027639 Order of unitary 2^n X 2^n group H_{n,4} acting on Siegel modular forms.

Original entry on oeis.org

4, 32, 3072, 2752512, 21139292160, 1342091380654080, 692647993190048071680, 2882479558988139892026900480, 96342151992701835341576224427212800, 25811138467998276182105365247324712232550400

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..45
Bernhard Runge, On Siegel modular forms. Part I, J. Reine Angew. Math., 436 (1993), 57-85.
Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
Index entries for sequences related to modular groups.

Cf. A005329, A048651.

A027639:= func< n | n eq 0 select 4 else 2^(n^2+2*n+2)*(&*[2^j-1: j in [1..n]]) >;
[A027639(n): n in [0..15]]; // G. C. Greubel, Aug 04 2022

seq(2^(n^2+2*n+2)*mul(2^i-1, i=1..n), n=0..10);

a[n_]:= (-1)^n*2^(n^2 +2*n+2)*QPochhammer[2,2,n];
Table[a[n], {n, 0, 15}] (* G. C. Greubel, Aug 04 2022 *)

a(n) = my(ret=1); for(i=1,n, ret = ret<Kevin Ryde, Aug 13 2022

from sage.combinat.q_analogues import q_pochhammer
def A027639(n): return (-1)^n*2^(n^2 + 2*n + 2)*q_pochhammer(n, 2, 2)
[A027639(n) for n in (0..15)] # G. C. Greubel, Aug 04 2022

a(n) = 2^(n^2 + 2*n + 2) * Product_{j=1..n} (2^j - 1).
a(n) = (-1)^n * 2^(n^2 + 2*n + 2) * (2, 2){n}, where (q, q){n} is the q-Pochhammer symbol. - G. C. Greubel, Aug 04 2022
a(n) ~ c * 2^((3*n^2+5*n+4)/2), where c = A048651. - Amiram Eldar, Jul 09 2025

A123672 a(1) = 1; for n > 1, a(n) = (2^n-1)*a(n-1) + (-1)^n.

Original entry on oeis.org

1, 4, 27, 406, 12585, 792856, 100692711, 25676641306, 13120763707365, 13422541272634396, 27475941985082608611, 112513982428913282262046, 921602030075228695008418785, 15098606058722471710322924954656, 494736024726159230532151281989213151

Offset: 1

Benoit Cloitre, Nov 16 2006

nonn

This sequence allows us to prove that the constant C defined in A048651 is irrational. Indeed, for any n > 1 we get |(C+1)*A005329(n) - a(n)| < 1/2^n.

G. C. Greubel, Table of n, a(n) for n = 1..80
J. Lynch et al., Problem 6233: Irrationality of an infinite product, Amer. Math. Monthly 87 (1980) 408-409

Cf. A005329, A048651, A277357.

[n eq 1 select 1 else (2^n-1)*Self(n-1)+(-1)^n: n in [1..15]]; // Vincenzo Librandi, Oct 18 2017

RecurrenceTable[{a[n] == (2^n - 1) * a[n - 1] + (-1)^n, a[1] == 1}, a, {n, 1, 15}] (* Vaclav Kotesovec, Oct 10 2016 *)

PARI
```
a(n)=if(n<2,1,(2^n-1)*a(n-1)+(-1)^n)
```

a(n) ~ c * 2^(n*(n+1)/2), where c = 0.372186658950350942813441530084543367... . - Vaclav Kotesovec, Oct 10 2016

A182263 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * d^n/dx^n x*A(x)^2.

Original entry on oeis.org

1, 1, 6, 91, 2910, 187178, 24019884, 6154080275, 3151538898870, 3227331249742334, 6609648919647088788, 27073195436180090799006, 221783764770326660974008300, 3633705802215756626623500731892, 119069276624759801067298501607804376

Offset: 0

Paul D. Hanna, Apr 21 2012

nonn

			G.f.: A(x) = 1 + x + 6*x^2 + 91*x^3 + 2910*x^4 + 187178*x^5 + 24019884*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 13*x^2 + 194*x^3 + 6038*x^4 + 381268*x^5 + 48457325*x^6 + 12358976074*x^7 + 6315716731394*x^8 + 6461044887240556*x^9 +...
such that a(n) = (2^n-1) times the coefficient of x^(n-1) in A(x)^2:
a(2) = 3 * 2 = 6;
a(3) = 7 * 13 = 91;
a(4) = 15 * 194 = 2910;
a(5) = 31 * 6038 = 187178;
a(6) = 63 * 381268 = 24019884; ...

Vincenzo Librandi, Table of n, a(n) for n = 0..70

Cf. A005329, A182264.

a = ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = (2^n-1)* Sum[a[[k+1]]*a[[n-k]],{k,0,n-1}],{n,2,20}]; a  (* Vaclav Kotesovec, Feb 22 2014 *)

/* Generating Function Satisfies: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} /* = n-th derivative of F */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(k=1, n, x^k/k!*Dx(k, x*A^2+x*O(x^n) ))); polcoeff(A, n)}

/* Recurrence: */
{a(n)=if(n==0,1,(2^n-1)*sum(k=0,n-1,a(k)*a(n-k-1)))}
for(n=0,15,print1(a(n),", "))

/* Recurrence: */
{a(n)=local(A=1+sum(k=1,n-1,a(k)*x^k)+x*O(x^n));if(n==0,1,(2^n-1)*polcoeff(A^2,n-1))}

a(n) = (2^n-1) * { [x^(n-1)] A(x)^2 } for n>0 with a(0)=1.
a(n) = (2^n-1) * Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1.
a(n) ~ c * 2^((n-1)*(n+4)/2), where c = 0.71662215139236633556752111264619992099204134882... - Vaclav Kotesovec, Feb 22 2014

A216206 a(n) = Product_{i=1..n} ((-2)^i-1).

Original entry on oeis.org

1, -3, -9, 81, 1215, -40095, -2525985, 325852065, 83092276575, -42626337882975, -43606743654283425, 89350217747626737825, 365889141676531491393375, -2997729737755822508985921375, -49111806293653640164716349886625, 1609344780436736134557590069434814625

Offset: 0

R. J. Mathar, Mar 12 2013

Signed partial products of A062510. This implies that all terms from a(1) on are multiples of 3.

Cf. A005329, A015013, A015109, A027871, A062510, A330862.

A216206 := proc(n)
        mul( (-2)^i-1, i=1..n) ;
end proc:

Table[(-1)^n QPochhammer[-2, -2, n], {n, 0, 15}] (* Bruno Berselli, Mar 13 2013 *)
Table[Product[(-2)^k-1,{k,n}],{n,0,20}] (* Harvey P. Dale, Oct 21 2024 *)

A015109(n,k) = a(n)/(a(k)*a(n-k)).
a(n) = (-3)^n*A015013(n) for n>0, a(0)=1. - Bruno Berselli and Alonso del Arte, Mar 13 2013
a(n) ~ (-1)^(floor(n/2)+1) * c * 2^(n*(n+1)/2), where c = Product_{k>=1} (1 - 1/(-2)^k) = 1.21072413030105918013... (A330862). - Amiram Eldar, Aug 10 2025

A259970 Triangle read by rows: coefficients eta(n,k) arising from the study of completely transitive graphs on n nodes.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 6, 28, 42, 21, 91, 510, 1050, 945, 315, 2820, 18631, 48360, 61845, 39060, 9765, 177661, 1351413, 4220433, 6942915, 6357015, 3075975, 615195

Offset: 1

N. J. A. Sloane, Jul 11 2015

			Triangle begins:
1,
0,1,
1,3,3,
6,28,42,21,
91,510,1050,945,315,
2820,18631,48360,61845,39060,9765,
177661,1351413,4220433,6942915,6357015,3075975,615195,
...

E. Andresen, K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
Hsien-Kuei Hwang, Emma Yu Jin, Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], Dec 25 2020, p. 29.

Cf. A259971, A259972.
Diagonals include A005327, A005328, A005329.
Row sums are A005014.

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

A243950 Sum of the squares of q-binomial coefficients for q=2 in row n of triangle A022166, for n >= 0.

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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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A335011 Decimal expansion of Product_{k>=1} (1 - 1/2^k)^k.

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A027639 Order of unitary 2^n X 2^n group H_{n,4} acting on Siegel modular forms.

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A123672 a(1) = 1; for n > 1, a(n) = (2^n-1)*a(n-1) + (-1)^n.

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A182263 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * d^n/dx^n x*A(x)^2.

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