A155103 -id:A155103 - cp's OEIS frontend

A155105 Positive numbers appearing in the third column of A155103.

1, 4, 28, 364, 9100, 445900, 43252300, 8347693900, 3213862151500, 2471459994503500, 3798634011551879500, 11673202317498925703500

Offset: 1

Mats Granvik, Jan 20 2009

Cf. A155103.

A155102 := proc(n,k) if n = k then 1 ; elif n =2*k then -k-1 ; else 0; end if; end proc:
A155103 := proc(amx) a := array(1..amx,1..amx) ; a[1,1] := 1/A155102(1,1) ;
        for r from 1 to amx do
                for c from 1 to r-1 do a[c,r] := 0 ; end do:
                a[r,r] := 1/A155102(r,r) ;
                for c from r-1 to 1 by -1 do a[r,c] := -add(a[cp,c]*A155102(r,cp),cp=c..r-1)/A155102(r,r) ;
                        if c = 3 and a[r,c] <> 0 then print( a[r,c]) ; end if;
                end do:
        end do:
        return ;
end proc:
A155103(290) ; # R. J. Mathar, Dec 07 2010

\\ after R. J. Mathar
T(n,k)=if(n==k,1,if(n==2*k,-(k+1))); \\ from A155102
\\ First term = 1 omitted
a155103(upto) = my(m=3*2^upto, a=matid(m)); for(r=1, m, forstep(c=r-1, 1, -1, a[r,c]=-sum(cp=c, r-1, a[cp,c]*T(r,cp)); if(c==3 && a[r,c]!=0, print1(a[r,c],", "))));
a155103(8) \\ Hugo Pfoertner, Oct 03 2024

From Tristan Cam, Oct 02 2024: (Start)
a(1) = 1, a(n) = a(n-1)*(1+3*2^(n-2)) (conjectured).
a(n) = Product_{k=1..n-1} 1+3*2^(k-1) = QPochhammer[-3, 2, n-1]. (conjectured). (End)

Two more terms from R. J. Mathar, Dec 07 2010

a(8)-a(12) from Hugo Pfoertner, Oct 02 2024

A155104 Numbers appearing in the fourth column of A155103.

Original entry on oeis.org

1, 5, 45, 765, 25245, 1640925, 211679325, 54401586525, 27908013887325, 28605714234508125, 58613108466507148125, 240137905387279785868125, 1967449858837983285617548125, 32236665937060356134843526028125

Offset: 1

Mats Granvik, Jan 20 2009

nonn

			a(1)=1, a(2)=1*5, a(3)=1*5*9, a(4)=1*5*9*17, a(5)=1*5*9*17*33, a(6)=1*5*9*17*33*65 and so on. See A000051 for continuation.

Cf. A000051, A155105.

Table[QPochhammer[-4, 2, n - 1], {n, 14}] (* Arkadiusz Wesolowski, Oct 29 2012 *)

a(n) = prod(k=2..n) (1+2^k), n>1. - Mats Granvik, Simone Severini, Dec 04 2010

a(n) = A028362(n+1)/3. - Vladimir Reshetnikov, Sep 16 2016

More terms from Mats Granvik, Dec 05 2010

A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

Original entry on oeis.org

1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375

Offset: 1

N. J. A. Sloane

These numbers appear in the second column of A155103. - Mats Granvik, Jan 20 2009

a(n) = n terms in the sequence (1, 2, 4, 8, 16, ...) dot n terms in the sequence (1, 1, 3, 15, 135). Example: a(5) = 2295 = (1, 2, 4, 8, 16) dot (1, 1, 3, 15, 135) = (1 + 2 + 12 + 120 + 2160). - Gary W. Adamson, Aug 02 2010

			G.f. = x + 3*x^2 + 15*x^3 + 135*x^4 + 2295*x^5 + 75735*x^6 + 4922775*x^7 + ...

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 630.

T. D. Noe, Table of n, a(n) for n = 1..50
C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
Steven T. Dougherty and Esengül Saltürk, The neighbor graph of binary self-orthogonal codes, Adv. Math. Comm. (2024). See p. 6.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

Cf. A000124, A003178, A003179, A028361, A028363, A048675, A053632, A068052 (XOR-analog), A285101.
Cf. A155103. - Mats Granvik, Jan 20 2009
Cf. A005329, A006088. - Paul D. Hanna, Sep 16 2009

[1] cat [&*[ 2^k+1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015

seq(mul(1 + 2^j, j = 1..n-1), n = 1..20); # G. C. Greubel, Jun 06 2020

Table[Product[2^i+1,{i,n-1}],{n,15}] (* or *) FoldList[Times,1, 2^Range[15]+1] (* Harvey P. Dale, Nov 21 2011 *)
Table[QPochhammer[-2, 2, n - 1], {n, 15}] (* Arkadiusz Wesolowski, Oct 29 2012 *)

{a(n)=polcoeff(sum(m=0,n,2^(m*(m-1)/2)*x^m/prod(k=0,m-1,1-2^k*x+x*O(x^n))),n)} \\ Paul D. Hanna, Sep 16 2009

{a(n) = if( n<1, 0 , prod(k=1, n-1, 2^k + 1))}; /* Michael Somos, Jan 28 2018 */

{a(n) = sum(k=0, n-1, 2^(k*(k+1)/2) * prod(j=1, k, (2^(n-j) - 1) / (2^j - 1)))}; /* Michael Somos, Jan 28 2018 */

for n in range(2,40,2):
  product = 1
  for i in range(1,n//2-1 + 1):
    product *= (2**i+1)
  print(product)
# Nathan J. Russell, Mar 01 2016

from math import prod
def A028362(n): return prod((1<Chai Wah Wu, Jun 20 2022

from ore_algebra import *
R. = QQ['x']
A. = OreAlgebra(R, 'Qx', q=2)
print((Qx - x - 1).to_list([0,1], 10))  # Ralf Stephan, Apr 24 2014

from sage.combinat.q_analogues import q_pochhammer
[q_pochhammer(n-1,-2,2) for n in (1..20)] # G. C. Greubel, Jun 06 2020

;; With memoization-macro definec.
(define (A028362 n) (A028362off0 (- n 1)))
(definec (A028362off0 n) (if (zero? n) 1 (+ (A028362off0 (- n 1)) (* (expt 2 n) (A028362off0 (- n 1))))))
;; Antti Karttunen, Apr 15 2017

a(n) = Product_{i=1..n-1} (2^i+1).
Letting a(0)=1, we have a(n) = Sum_{k=0..n-1} 2^k*a(k) for n>0. a(n) is asymptotic to c*sqrt(2)^(n^2-n) where c=2.384231029031371724149899288.... = A079555 = Product_{k>=1} (1 + 1/2^k). - Benoit Cloitre, Jan 25 2003
G.f.: Sum_{n>=1} 2^(n*(n-1)/2) * x^n/(Product_{k=0..n-1} (1-2^k*x)). - Paul D. Hanna, Sep 16 2009
a(n) = 2^(binomial(n,2) - 1)*(-1; 1/2){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
From Antti Karttunen, Apr 15 2017: (Start)
a(n) = A048675(A285101(n-1)).
a(n) = b(n-1), where b(0) = 1, and for n > 0, b(n) = b(n-1) + (2^n)*b(n-1).
a(n) = Sum_{i=1..A000124(n-1)} A053632(n-1,i-1)*(2^(i-1)) [where the indexing of both rows and columns of irregular table A053632(row,col) is considered to start from zero].
(End)
G.f. A(x) satisfies: A(x) = x * (1 + A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020
Conjectural o.g.f. as a continued fraction of Stieltjes type (S-fraction):
  1/(1 - 3*x/(1 - 2*x/(1 - 10*x/(1 - 12*x/(1 - 36*x/(1 - 56*x/(1 - 136*x/(1 - 240*x/(1 - ... - 2^(n-1)*(2^n + 1)*x/(1 - 2^n*(2^n - 1)*x/(1 - ... ))))))))))). - Peter Bala, Sep 27 2023

A028361 Number of totally isotropic spaces of index n in orthogonal geometry of dimension 2n.

Original entry on oeis.org

1, 2, 6, 30, 270, 4590, 151470, 9845550, 1270075950, 326409519150, 167448083323950, 171634285407048750, 351678650799042888750, 1440827432323678715208750, 11804699153027899713705288750, 193419995622362136809061156168750, 6338179836549184861096125026493768750

Offset: 0

N. J. A. Sloane

These numbers appear in first column of A155103. - Mats Granvik, Jan 20 2009
Equals row sums of unsigned triangle A158474. - Gary W. Adamson, Mar 20 2009
a(n) = (n+2) terms in the sequence (1, 1, 2, 4, 8, 16, ...) dot (n+2) terms in the sequence (1, 1, 2, 6, 30, 270, ...). Example: a(4) = 4590 = (1, 2, 4, 8, 16) dot (1, 1, 2, 6, 30, 270) = (1 + 1 + 4 + 24 + 240 + 4320). - Gary W. Adamson, Aug 02 2010
a(n) is the right-hand side of the mass formula used to classify Type II Self Dual Binary Linear Codes of length 2(n+1). a(n) is the number of distinct Type II Self Dual Binary Linear codes of length 2(n+1) when 2(n+1) = 0 MOD 8.  It is important to note that Type II codes are only possible when the length is a multiple of 8. In short, this sequence only applies to Type II codes when 2(n+1) = 0 MOD 8, else the right hand side of the mass formula is zero. - Nathan J. Russell, Mar 04 2016
This is almost certainly the sequence of number of Carlyle circles needed for the construction of regular polygons using straightedge and compass mentioned on page 107 of DeTemple (1991). - N. J. A. Sloane, Aug 05 2021
a(n) is also the number of Sp(oo, F2)-orbits of V^n, where V is the countable-dimensional symplectic vector space over the two-element field. - Jingjie Yang, Jul 30 2025

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Page 366. - Nathan J. Russell, Mar 04 2016

T. D. Noe, Table of n, a(n) for n=0..50
C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
Julien Clément and Antoine Genitrini, Binary Decision Diagrams: from Tree Compaction to Sampling, arXiv:1907.06743 [cs.DS], 2019.
John P. D'Angelo, Counting Tournament Brackets, J. Int. Seq., Vol. 25 (2022), Article 22.6.8.
Duane W. DeTemple, Carlyle circles and the Lemoine simplicity of polygon constructions, The American Mathematical Monthly 98.2 (1991): 97-108.
Davide Gaiotto and Justin Kulp, Orbifold groupoids, arXiv:2008.05960 [hep-th], 2020, page 42.

Cf. A006125, A028362, A155103, A158474, A323716 (product of 3^i+1).

[1] cat [ (&*[1+2^j: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Jun 06 2020

seq( mul((1+2^j), j=0..n-1), n = 0..20); # G. C. Greubel, Jun 06 2020

Table[QPochhammer[-1, 2, n], {n, 0, 15}] (* Arkadiusz Wesolowski, Oct 29 2012 *)
Table[Product[2^i + 1, {i, 0, n/2 - 2}], {n, 2, 32, 2}] (* Nathan J. Russell, Mar 04 2016 *)
Table[Product[2^i + 1, {i, 0, n - 1}], {n, 0, 15}] (* Nathan J. Russell, Mar 04 2016 *)
FoldList[Times,1,2^Range[0,20]+1] (* Harvey P. Dale, Apr 11 2016 *)

PARI
```
{a(n) = prod(k=0, n-1, 2^k + 1)};
```

{a(n)=polcoeff(sum(m=0,n,2^(m*(m-1)/2)*x^m/prod(k=0,m,1-2^k*x+x*O(x^n))),n)} /* Paul D. Hanna, May 02 2012 */

for n in range(2,50,2):
  product = 1
  for i in range(0,n//2-2 + 1):
    product *= (2**i+1)
  print(product)
# Nathan J. Russell, Mar 01 2016

[product( 1+2^j for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Jun 06 2020

a(n) = Product_{i=0..n-1} ( 2^i + 1 ).
Asymptotic to C*2^(n*(n-1)/2) where C = A081845 = 4.76846205806274344829979857... = Product_{k>=0} (1 + 1/2^k). - Benoit Cloitre, Apr 09 2003
It appears that a(n) = 2^((1/2)*(n - 1)*n) * Product_{k>=0} (1 + 1/(2^k)) / Product_{k>=0} (1 + 1/(2^(n + k))). - Peter Moxey (pmoxey(AT)live.com), Mar 21 2010
G.f.: Sum_{n>=0} 2^(n*(n-1)/2) * x^n / Product_{k=0..n} (1 - 2^k*x). - Paul D. Hanna, May 02 2012
a(n) = (a(n-2)^3 + a(n-1) * a(n-3) * (a(n-1) - 2 * a(n-2))) * a(n-1) / (a(n-2)^2 * (a(n-2) - a(n-3))) if n>2. - Michael Somos, Aug 21 2012
0 = a(n)*(+a(n+1) + a(n+2)) + a(n+1)*(-2*a(n+1)) for all n>=0. - Michael Somos, Oct 10 2014
Sum_{k=0..n} 2^k/a(k) = 3-2/a(n) and Sum_{k=0..n} 4^k/a(k) = 9-(4*(1+2^n))/a(n) for n >= 0. - Werner Schulte, Dec 25 2016
G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020
a(n) = Sum_{k=0..n} q_binomial(n, k, q=2) * 2^(k*(k-1)/2). - Jingjie Yang, Jul 30 2025

A155102 Triangle T(n,k) read by rows. If n=k then T(n,k)=1, elseif n=2*k then T(n,k)=-(k+1), else T(n,k)=0.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Jan 20 2009

Matrix inverse of this triangle is A155103.

			Table begins:
1,
-2,1,
0,0,1,
0,-3,0,1,
0,0,0,0,1,
0,0,-4,0,0,1,
0,0,0,0,0,0,1,
0,0,0,-5,0,0,0,1,

Cf. A155103.

=if(row()=column();1;if(and(row()>=column()*2;row()<=column()*2);-1;0))*(row()-column()+1)

t[n_, k_] := Which[n == k, 1, n == 2k, -k - 1, True, 0]; Flatten[ Table[t[n, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, Jul 19 2012 *)

T(n,k)=if(n==k,1,if(n==2*k,-(k+1))); for(n=1,20,for(k=1,n,print1(T(n,k),", "));print())

Definition clarified by Mats Granvik, Dec 05 2010

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A155105 Positive numbers appearing in the third column of A155103.

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A155104 Numbers appearing in the fourth column of A155103.

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A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

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A028361 Number of totally isotropic spaces of index n in orthogonal geometry of dimension 2n.

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A155102 Triangle T(n,k) read by rows. If n=k then T(n,k)=1, elseif n=2*k then T(n,k)=-(k+1), else T(n,k)=0.

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