cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A005329 a(n) = Product_{i=1..n} (2^i - 1). Also called 2-factorial numbers.

Original entry on oeis.org

1, 1, 3, 21, 315, 9765, 615195, 78129765, 19923090075, 10180699028325, 10414855105976475, 21319208401933844325, 87302158405919092510875, 715091979502883286756577125, 11715351900195736886933003038875, 383876935713713710574133710574817125
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Conjecture: this sequence is the inverse binomial transform of A075272 or, equivalently, the inverse binomial transform of the BinomialMean transform of A075271. - John W. Layman, Sep 12 2002
To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125. The probability of having won before n+1 tails is A114604 / A006125. - Joshua Zucker, Dec 14 2005
Number of upper triangular n X n (0,1)-matrices with no zero rows. - Vladeta Jovovic, Mar 10 2008
Equals the q-Fibonacci series for q = (-2), and the series prefaced with a 1: (1, 1, 1, 3, 21, ...) dot (1, -2, 4, -8, ...) if n is even, and (-1, 2, -4, 8, ...) if n is odd. For example, a(3) = 21 = (1, 1, 1, 3) dot (-1, 2, -4, 8) = (-1, 2, -4, 24) and a(4) = 315 = (1, 1, 1, 3, 21) dot (1, -2, 4, -8 16) = (1, -2, 4, -24, 336). - Gary W. Adamson, Apr 17 2009
Number of chambers in an A_n(K) building where K=GF(2) is the field of two elements. This is also the number of maximal flags in an n-dimensional vector space over a field of two elements. - Marcos Spreafico, Mar 22 2012
Given probability p = 1/2^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, A114604(n)/A006125(n+2) = 1-a(n)/A006125(n+1) is the probability that the outcome has occurred up to and including the n-th iteration. The limiting ratio is 1-A048651 ~ 0.7112119. These observations are a more formal and generalized statement of Joshua Zucker's Dec 14, 2005 comment. - Bob Selcoe, Mar 02 2016
Also the number of dominating sets in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Empirical: Letting Q denote the Hall-Littlewood Q basis of the symmetric functions over the field of fractions of the univariate polynomial ring in t over the field of rational numbers, and letting h denote the complete homogeneous basis, a(n) is equal to the absolute value of 2^A000292(n) times the coefficient of h_{1^(n*(n+1)/2)} in Q_{(n, n-1, ..., 1)} with t evaluated at 1/2. - John M. Campbell, Apr 30 2018
The series f(x) = Sum_{n>=0} x^(2^n-1)/a(n) satisfies f'(x) = f(x^2), f(0) = 1. - Lucas Larsen, Jan 05 2022

Examples

			G.f. = 1 + x + 3*x^2 + 21*x^3 + 315*x^4 + 9765*x^5 + 615195*x^6 + 78129765*x^7 + ...

References

  • Annie Cuyt, Vigdis Brevik Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones, Handbook of continued fractions for special functions, Springer, New York, 2008. (see 19.2.1)
  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 358.
  • Mark Ronan, Lectures on Buildings (Perspectives in Mathematics; Vol. 7), Academic Press Inc., 1989.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000225, A005321, A006125, A114604, A006088, A028362, A027871 (3-fac), A027872 (5-fac), A027873 (6-fac), A048651, A048652, A075271, A075272, A032085, A122746.
Cf. A048651, A079555, A152476 (inverse binomial transform).
Column q=2 of A069777.

Programs

  • GAP
    List([0..15],n->Product([1..n],i->2^i-1)); # Muniru A Asiru, May 18 2018
  • Magma
    [1] cat [&*[ 2^k-1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015
  • Maple
    A005329 := proc(n) option remember; if n<=1 then 1 else (2^n-1)*procname(n-1); end if; end proc: seq(A005329(n), n=0..15);
  • Mathematica
    a[0] = 1; a[n_] := a[n] = (2^n-1)*a[n-1]; a /@ Range[0,14] (* Jean-François Alcover, Apr 22 2011 *)
FoldList[Times, 1, 2^Range[15] - 1] (* Harvey P. Dale, Dec 21 2011 *)
Table[QFactorial[n, 2], {n, 0, 14}] (* Arkadiusz Wesolowski, Oct 30 2012 *)
QFactorial[Range[0, 10], 2] (* Eric W. Weisstein, Jul 14 2017 *)
a[ n_] := If[ n < 0, 0, (-1)^n QPochhammer[ 2, 2, n]]; (* Michael Somos, Jan 28 2018 *)
  • PARI
    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, Sep 17 2009
  • PARI
    Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D
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+x*O(x^n) ))); polcoeff(A,n) \\ Paul D. Hanna, Apr 21 2012
  • PARI
    {a(n) = if( n<0, 0, prod(k=1, n, 2^k - 1))}; /* Michael Somos, Jan 28 2018 */
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sum(k=0, n+1, (-1)^k * 2^(k*(k+1)/2) * prod(j=1, k, (2^(n+1-j) - 1) / (2^j - 1))))}; /* Michael Somos, Jan 28 2018 */

Formula

a(n)/2^(n*(n+1)/2) -> c = 0.2887880950866024212788997219294585937270... (see A048651, A048652).
From Paul D. Hanna, Sep 17 2009: (Start)
G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n / (Product_{k=0..n} (1+2^k*x)).
Compare to: 1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/(Product_{k=1..n+1} (1+2^k*x)). (End)
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * d^n/dx^n x*A(x). - Paul D. Hanna, Apr 21 2012
a(n) = 2^(binomial(n+1,2))*(1/2; 1/2){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
a(n) = Product_{i=1..n} A000225(i). - Michel Marcus, Dec 27 2015
From Peter Bala, Nov 10 2017: (Start)
O.g.f. as a continued fraction of Stieltjes' type: A(x) = 1/(1 - x/(1 - 2*x/(1 - 6*x/(1 - 12*x/(1 - 28*x/(1 - 56*x/(1 - ... -(2^n - 2^floor(n/2))*x/(1 - ... )))))))) (follows from Heine's continued fraction for the ratio of two q-hypergeometric series at q = 2. See Cuyt et al. 19.2.1).
A(x) = 1/(1 + x - 2*x/(1 - (2 - 1)^2*x/(1 + x - 2^3*x/(1 - (2^2 - 1)^2*x/(1 + x - 2^5*x/(1 - (2^3 - 1)^2*x/(1 + x - 2^7*x/(1 - (2^4 - 1)^2*x/(1 + x - ... ))))))))). (End)
0 = a(n)*(a(n+1) - a(n+2)) + 2*a(n+1)^2 for all n>=0. - Michael Somos, Feb 23 2019
From Amiram Eldar, Feb 19 2022: (Start)
Sum_{n>=0} 1/a(n) = A079555.
Sum_{n>=0} (-1)^n/a(n) = A048651. (End)

Extensions

Better definition from Leslie Ann Goldberg (leslie(AT)dcs.warwick.ac.uk), Dec 11 1999

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

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

References

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

Links

Crossrefs

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

Programs

  • Magma
    [1] cat [&*[ 2^k+1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015
  • Maple
    seq(mul(1 + 2^j, j = 1..n-1), n = 1..20); # G. C. Greubel, Jun 06 2020
  • Mathematica
    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 *)
  • PARI
    {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
  • PARI
    {a(n) = if( n<1, 0 , prod(k=1, n-1, 2^k + 1))}; /* Michael Somos, Jan 28 2018 */
  • PARI
    {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 */
  • Python
    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
  • Python
    from math import prod
def A028362(n): return prod((1<Chai Wah Wu, Jun 20 2022
  • Sage
    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
  • Sage
    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
  • Scheme
    ;; 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

Formula

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

A126391 a(1)=1; for n>1: a(n) = sum of all subsets of (a(1),..,a(n-1)).

Original entry on oeis.org

1, 1, 4, 24, 240, 4320, 146880, 9694080, 1260230400, 325139443200, 167121673804800, 171466837323724800, 351507016513635840000, 1440475753672879672320000, 11803258325595576034990080000
Offset: 1

Views

Author

Zak Seidov, Mar 23 2007

Keywords

Comments

Is connection with A006088 clear?
Apparently first differences of A028361. - Sean A. Irvine, Jun 25 2022

Examples

			n=2: subsets of (1) are ((0),(1)), sums of subsets are (0,1) and total sum is 0+1=1, hence a(2)=1;
n=3: subsets of (1,1) are ((0),(1),(1),(1,1)), sums of subsets are (0,1,1,2) and total sum is 0+1+1+2=4, hence a(3)=4;
n=4: subsets of (1,1,4) are ((0),(1),(1),(4),(1,1),(1,4),(1,4),(1,1,4)), sums of subsets are (0,1,1,4,2,5,5,6) and total sum is 0+1+1+4+2+5+5+6=24, hence a(4)=24.

Crossrefs

Cf. A006088.

Programs

  • Mathematica
    a[1]=1;a[2]=1;a[n_]:=a[n]=(2^(n-2)+2)*a[n-1];Table[a[i],{i,18}]

Formula

a(1)=1, a(2)=1; n>2: a(n)=(2^(n-2)+2)*a(n-1). a(1)=1; n>1: a(n)=A006088(n-1).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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