A028362 -id:A028362 - cp's OEIS frontend

A285101 a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)), where A242378(n,a(n-1)) shifts the prime factorization of a(n-1) n primes towards larger primes with A003961.

2, 6, 210, 3573570, 64845819350301990, 28695662573739152697846686144187168109530, 1038300112150956151877699324649731518883355380534272386781875587619359740733888844803014212990

Offset: 0

Antti Karttunen, Apr 15 2017

nonn

Multiplicative encoding of irregular table A053632 (in style of A007188 and A260443).

Indranil Ghosh, Table of n, a(n) for n = 0..9

Cf. A001222, A003961, A007913, A028362, A048675, A053632, A068052, A242378, A248663, A285102.

Cf. also A007188, A260443.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A242378(k,n) = { while(k>0,n = A003961(n); k = k-1); n; };
A285101(n) = { if(0==n,2,A285101(n-1)*A242378(n,A285101(n-1))); };

from sympy import factorint, prime, primepi
from operator import mul
from functools import reduce
def a003961(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
def a242378(k, n):
    while k>0:
        n=a003961(n)
        k-=1
    return n
l=[2]
for n in range(1, 7):
    x=l[n - 1]
    l.append(x*a242378(n, x))
print(l) # Indranil Ghosh, Jun 27 2017

(definec (A285101 n) (if (zero? n) 2 (* (A285101 (- n 1)) (A242378bi n (A285101 (- n 1)))))) ;; For A242378bi see A242378.

a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)).
Other identities. For all n >= 0:
A001222(a(n)) = A000079(n).
A048675(a(n)) = A028362(1+n).
A248663(a(n)) = A068052(n).
A007913(a(n)) = A285102(n).

A155103 Triangle read by rows: Matrix inverse of A155102.

Original entry on oeis.org

1, 2, 1, 0, 0, 1, 6, 3, 0, 1, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 30, 15, 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, 28, 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, 0, 0

Offset: 1

Mats Granvik, Jan 20 2009

A028361 appears in the first column at A036987 positions. A028362 appears in the second column, A155105 in the third and A155104 in the fourth. A000384 appears as the third ray from zero and A100147 as the fourth.

			Table begins:
1,
2,1,
0,0,1,
6,3,0,1,
0,0,0,0,1,
0,0,4,0,0,1,
0,0,0,0,0,0,1,
30,15,0,5,0,0,0,1,

Cf. A028361, A036987, A028362, A155105, A155104. A000384, A100147.

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

A269455 Number of Type I (singly-even) self-dual binary codes of length 2n.

Original entry on oeis.org

1, 3, 15, 105, 2295, 75735, 4922775, 625192425, 163204759575, 83724041661975, 85817142703524375, 175667691114114395625, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375, 3168896498278970068411253452090715625, 207692645973961964120828372930661061284375, 27222898185745116523209337325140537285726884375, 7136346644902153570976711733098966146766874104484375, 3741493773415815389266667264411257664189964123617799515625

Offset: 1

Nathan J. Russell, Feb 27 2016

nonn

A self dual binary linear code is either Type I (singly even) or Type II (doubly even). A self dual binary linear code can only be Type II if the length of the code (2n) is a multiple of 8. The total number self dual binary linear codes (including equivalent codes) is equal to the number of Type I self dual binary linear codes (including equivalent codes) when the length (2n) is not a multiple of 8. If the length is a multiple of 8 ( 2n =0 mod 8 ) then the total number of Type I codes is the number of type II codes subtracted from the total number of self dual codes of length 2n.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, Page 366.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

Nathan J. Russell, Table of n, a(n) for n = 1..49
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
P. Gaborit, Tables of Self-Dual Codes
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Cf. A003178, A003179, A028363, A028361.

Table[
If[Mod[2 n, 8] == 0,
  Product[2^i + 1, {i, 1, n - 1}] - Product[2^i + 1, {i, 0, n - 2}] ,
  Product[2^i + 1, {i, 1, n - 1}]],
{n, 1, 10}] (* Nathan J. Russell, Mar 01 2016 *)

a(n) = if (2*n%8==0, prod(i=1, n-1, 2^i+1)-prod(i=0, n-2, 2^i+1), prod(i=1, n-1, 2^i+1))
vector(20, n, a(n)) \\ Colin Barker, Feb 28 2016

for n in range(1,10):
    product1 = 1
    for i in range(1,n-1 + 1):
        product1 *= (2**i+1)
    if (2*n)%8 == 0:
        product2 = 1
        for i in range(n-2 + 1):
            product2 *= (2**i+1)
        print(product1 - product2)
    else:
        print(product1)

From Nathan J. Russell, Mar 01 2016: (Start)
If 2n = 0 MOD 8 then a(n) = prod_(2^i+1, i=1,...,n-1) - prod_(2^i+1, i=0,...,n-2);
If 2n != 0 MOD 8 then a(n) = prod_(2^i+1, i=1,...,n-1).
If 2n = 0 MOD 8 then a(n) = A028362(n) - A028363( n/8);
If 2n != 0 MOD 8 then a(n) = A028362(n).
(End)

a(20) corrected by Andrew Howroyd, Feb 22 2018

A322299 Number of distinct automorphism group sizes for binary self-dual codes of length 2n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 24, 48, 85, 149, 245, 388

Offset: 1

Nathan J. Russell, Dec 02 2018

Codes are vector spaces with a metric defined on them.  Specifically, the metric is the hamming distance between two vectors.  Vectors of a code are called codewords.
A code is usually represented by a generating matrix.  The row space of the generating matrix is the code itself.
Self-dual codes are codes such all codewords are pairwise orthogonal to each other.
Two codes are called permutation equivalent if one code can be obtained by permuting the coordinates (columns) of the other code.
The automorphism group of a code is the set of permutations of the coordinates (columns) that result in the same identical code.

			There are a(16) = 388 distinct sizes for the automorphism groups of the binary self-dual codes of length 16.  In general, two automorphism  groups with the same size are not necessarily isomorphic.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, pp. 7, 252-330, 338-393.

Cf. self-dual codes A028362, A003179, A106162, A028363, A106163.

A006088 a(n) = (2^n + 2) a(n-1) (kissing number of Barnes-Wall lattice in dimension 2^n).

Original entry on oeis.org

1, 4, 24, 240, 4320, 146880, 9694080, 1260230400, 325139443200, 167121673804800, 171466837323724800, 351507016513635840000, 1440475753672879672320000, 11803258325595576034990080000, 193408190923209108909347450880000

Offset: 0

N. J. A. Sloane, John Leech

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 151.
Robert L. Griess Jr., Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices, arXiv:math/0403480 [math.GR], Mar 28 2004. See Proposition 8.9.
J. Leech, Some sphere packings in higher space, Canad. J. Math., 16 (1964), 657-682.
J. Leech & N. J. A. Sloane, Correspondence, 1975
C. Musès, The dimensional family approach in (hyper)sphere packing..., Applied Math. Computation 88 (1997), pp. 1-26, see p. 22.
G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
Index entries for sequences related to Barnes-Wall lattices

Cf. A028362. - Paul D. Hanna, Sep 16 2009

Cf. A081845.

I:=[4]; [1] cat [n le 1 select I[n] else (2^n + 2)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Dec 31 2015

a[0]:=1: for n from 1 to 16 do a[n]:=(2^n+2)*a[n-1] od: seq(a[n],n=0..16); # Emeric Deutsch, Dec 10 2004

RecurrenceTable[{a[0]==1, a[n]==(2^n + 2) a[n-1]}, a[n], {n, 0, 25}] (* Vincenzo Librandi, Dec 31 2015 *)

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

a(n) = prod(k=1, n, 2+2^k); \\ Michel Marcus, Jan 01 2016

a(n) = (2+2)(2+4)(2+8)(2+16)...(2+2^n).
From Paul D. Hanna, Sep 16 2009: (Start)
G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1-2^k*x)];
contrast with:
1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1+2^k*x)]. (End)
a(n) ~ c * 2^(n*(n+1)/2), where c = A081845. - Vaclav Kotesovec, Dec 31 2015

More terms from Emeric Deutsch, Dec 10 2004

Replaced arXiv URL with non-cached version - R. J. Mathar, Oct 23 2009

A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 64, 56, 14, 1, 1024, 960, 280, 30, 1, 32768, 31744, 9920, 1240, 62, 1, 2097152, 2064384, 666624, 89280, 5208, 126, 1, 268435456, 266338304, 87392256, 12094464, 755904, 21336, 254, 1, 68719476736, 68451041280, 22638755840, 3183575040, 205605888, 6217920, 86360, 510, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...] = A014236 (first zero omitted) DELTA A077957 where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006

			Triangle begins:
      1;
      2,     1;
      8,     6,    1;
     64,    56,   14,    1;
   1024,   960,  280,   30,  1;
  32768, 31744, 9920, 1240, 62, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A023531 (q=0), A007318 (q=1), this sequence (q=2), A173007 (q=3), A173008 (q=4).

Cf. A006125, A022166, A028362.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,2): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

T[n_, k_, q_]:= T[n,k,q]= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

Sum_{k=0..n} T(n, k) = A028362(n).
T(n,0) = 2^(n*(n+1)/2) = A006125(n+1). - Philippe Deléham, Nov 05 2006
T(n,k) = 2^binomial(n+1-k,2) * A022166(n,k) for 0 <= k <= n. - Werner Schulte, Mar 25 2019

A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

Original entry on oeis.org

1, 1, 4, 40, 1120, 91840, 22408960, 16358540800, 35792487270400, 234870301468364800, 4623187014103292723200, 272999193182799435304960000, 48361261073946554365403054080000, 25701205307660304745058529866383360000, 40976048450930207702360695570691784048640000

Offset: 0

Ilya Gutkovskiy, Jun 06 2020

nonn

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

Cf. A027871, A034472, A047656, A119600, A132324, A323716.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), this sequence (m=3), A309327 (m=4).

[n lt 3 select 1 else (&*[3^j +1: j in [1..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 21 2021

Table[Product[3^k + 1, {k, 1, n - 1}], {n, 0, 14}]

a(n) = prod(k=1, n-1, 3^k + 1); \\ Michel Marcus, Jun 06 2020

from sage.combinat.q_analogues import q_pochhammer
[1]+[3^(binomial(n,2))*q_pochhammer(n-1, -1/3, 1/3) for n in (1..20)] # G. C. Greubel, Feb 21 2021

G.f. A(x) satisfies: A(x) = 1 + x * A(3*x) / (1 - x).
G.f.: Sum_{k>=0} 3^(k*(k - 1)/2) * x^k / Product_{j=0..k-1} (1 - 3^j*x).
a(0) = 1; a(n) = Sum_{k=0..n-1} 3^k * a(k).
a(n) ~ c * 3^(n*(n - 1)/2), where c = Product_{k>=1} (1 + 1/3^k) = 1.564934018567011537938849... = A132324.
a(n) = 3^(binomial(n+1,2))*(-1/3;1/3){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021

A322339 Smallest automorphism group size for a binary self-dual code of length 2n.

Original entry on oeis.org

2, 8, 48, 384, 2688, 10752, 46080, 73728, 82944, 82944, 36864, 12288, 3072, 384, 30, 2, 1

Offset: 1

Nathan J. Russell, Dec 04 2018

nonn

A code is usually represented by a generating matrix. The row space of the generating matrix is the code itself.
Self-dual codes are codes such all codewords are pairwise orthogonal to each other.
Two codes are called permutation equivalent if one code can be obtained by permuting the coordinates (columns) of the other code.
The automorphism group of a code is the set of permutations of the coordinates (columns) that result in the same identical code.
The values in the sequence are not calculated lower bounds. For each n there exists a binary self-dual code of length 2n with an automorphism group of size a(n).
Binary self-dual codes have been classified (accounted for) up to a certain length. The classification process requires the automorphism group size be known for each code. There is a mass formula to calculate the number of distinct binary self-dual codes of a given length. Sequence A028362 gives this count. The automorphism group size allows researchers to calculate the number of codes that are permutationally equivalent to a code. Each new binary self-dual code C of length m that is discovered will account for m!/aut(C) codes in the total number calculated by the mass formula. Aut(C) represents the automorphism size of the code C. Sequence A003179 gives number of binary self-dual codes up to permutation equivalence.
There is a notable open problem in coding theory regarding binary self-dual codes. Does there exist a type II binary self-dual code of length 72 with minimum weight 16? The founder of OEIS N. J. A. Sloane posed the question in 1973. The question has been posed in several coding theory textbooks since 1973. There are even some rewards regarding the existence and nonexistence of the code. Some of the major work involved with researching the existence of the code has involved calculating possibilities for the automorphism group of the (72, 36, 16) type II binary self-dual code. The weight distribution for the code is listed as the finite sequence A120373. The current research demonstrates that the size of the automorphism group for this code is relatively small, perhaps even trivial with size 1. This sequence shows that as the length of a binary self-dual code grows the minimum size of the automorphism group grows up to a point, namely length 18. It would appear that a binary self-dual code of length 72 would no chance at having a small automorphism group size. However, after length 18 the minimum possible automorphism size stops increasing and starts declining all the way down to trivial a(17) = 1 for length 2*17=34. This demonstrates that a trivial or small sized automorphism group does not rule out the existence of the unknown type II (72, 36, 16) code.

			The smallest automorphism group size a binary self-dual code of length 2*16 = 32 is a(16) = 2.

N.J.A. Sloane, Is there a (72,36) d=16 self-dual code, IEEE Trans. Inform. Theory, 19 (1973), 251.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338-393.
Jon-Lark Kim, A Prize Problem In Coding Theory, University of Louisville

Cf. Self-Dual Codes A028362, A003179, A106162, A028363, A106163, A269455, A120373.

Cf. Self-Dual Code Automorphism Groups A322299.

A131791 Triangle read by rows of 2^n terms for n>=0: let S(n) denote the initial 2^n terms of the partial sums of row n; starting with a single '1' in row 0, generate row n+1 by concatenating S(n) with the terms of S(n) when read in reverse order.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 6, 5, 3, 1, 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1, 1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269, 270, 270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14, 5, 1, 1, 6, 20, 49, 99, 175, 280, 415

Offset: 0

Paul D. Hanna, Jul 15 2007

Row sums (and central terms) form A028361: Product_{i=0..n-1} (2^i + 1).
I'm interested in the graph of S(n). It appears to tend to a limit curve if scaled appropriately, e.g., scaled to fit a [0,1] box by f_n(x) = T(n,[x*2^n])/A028361(n-1). In this setup I think that the limit curve f(x) satisfies f(0)=0, f(1-x)=f(x), f(1/2)=1, f'(x)=2f(2x) for x<=1/2. Is this equation solvable? - Martin Fuller, Aug 31 2007
From N. J. A. Sloane, Nov 13 2018: (Start)
Kenyon (1992) defines p_n(x) (n >= 0) to be the polynomial
p_n(x) = (1+x)*(1+x+x^2)*(1+x+x^2+x^3+x^4)*...*(1+x+...+x^(2^n)).
He shows among many other things that the coefficient of x^(floor(c*2^(n+1))) in p_n(x), for c in [0,1], is given by
(f(c)+o(1))*p_n(1)/2^(n+1),
where f : R -> R is a nonzero C^1 function satisfying
(i) support(f) is a subset of [0,1],
(ii) f(x) = f(1-x), and
(iii) f'(x) = 4*f(2*x) for 0 <= x <= 1/2.
These three properties define f uniquely up to multiplication by a scalar. Also f is C^oo, is nowhere analytic on [0,1], and is a "bump function", since its graph looks like a "bump".
This provides a fairly complete answer to Martin Fuller's question above. (End)

			Triangle begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 5, 6, 6, 5, 3, 1;
1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1;
1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269, 270, 270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14, 5, 1; ...
ILLUSTRATION OF GENERATING METHOD.
From row 2: [1,2,2,1], take the partial sums: [1,3,5,6] and concatenate to this the terms in reverse order: [6,5,3,1] to obtain row 3: [1,3,5,6, 6,5,3,1].

Richard Kenyon, Infinite scaled convolutions, Preprint, 1992 (apparently unpublished)

Paul D. Hanna, Rows 0 to 11 of the triangle, flattened.
Julien Clément, Antoine Genitrini, Binary Decision Diagrams: from Tree Compaction to Sampling, arXiv:1907.06743 [cs.DS], 2019.

Cf. A131792 (main diagonal); A028361, A028362.

p[-1]:=1:
lprint(seriestolist(series(p[-1],x,0)));
p[0]:=(1-x^2)/(1-x):
lprint(seriestolist(series(p[0],x,2)));
for n from 1 to 4 do
p[n]:=p[n-1]*(1-x^(2^n+1))/(1-x);
lprint(seriestolist(series(p[n],x,2^(n+1))));
od: # N. J. A. Sloane, Nov 13 2018

T[n_, k_] := SeriesCoefficient[Product[(1-x^(2^j+1))/(1-x), {j, 0, n-1}], {x, 0, k}];
Table[T[n, k], {n, 0, 6}, {k, 0, 2^n-1}] // Flatten (* Jean-François Alcover, Oct 01 2019 *)

T(n,k)=local(A=[1],B=[1]);if(n==0,1,for(i=0,n-1, B=Vec(Ser(A)/(1-x));A=concat(B,Vec(Pol(B)+O(x^#B))));A[k+1])
for(n=0,6,for(k=0,2^n-1,print1(T(n,k),", "));print())

T(n,k)=polcoeff(prod(j=0,n-1,(1-x^(2^j+1))/(1-x)),k)
for(n=0,6,for(k=0,2^n-1,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Aug 09 2009

T(n, 2^(n-1)) = A028361(n-1) for n>=1.
T(n, 2^(n-2)) = A028362(n-1) for n>=2.
Sum_{k=0..2^n-1} (k+1)*T(n,k) = A028362(n+1) for n>=0.
G.f. of row n: Product_{j=0..n-1} (1 - x^(2^j+1))/(1-x). - Paul D. Hanna, Aug 09 2009

A309327 a(n) = Product_{k=1..n-1} (4^k + 1).

Original entry on oeis.org

1, 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, 1678532740564688125136703125, 1760070825503098980191468752703125, 7382273863761775568111978346806480703125, 123854010565759745011512941023673583762640703125

Offset: 0

Ilya Gutkovskiy, Jun 06 2020

nonn

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

Cf. A027637, A052539, A053763.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), A290000 (m=3), this sequence (m=4).

[n lt 2 select 1 else (&*[4^j +1: j in [1..n-1]]): n in [0..15]]; // G. C. Greubel, Feb 21 2021

Table[Product[4^k + 1, {k, 1, n - 1}], {n, 0, 13}]
Join[{1}, Table[4^(Binomial[n,2])*QPochhammer[-1/4, 1/4, n-1], {n,15}]] (* G. C. Greubel, Feb 21 2021 *)

a(n) = prod(k=1, n-1, 4^k + 1); \\ Michel Marcus, Jun 06 2020

from sage.combinat.q_analogues import q_pochhammer
[1]+[4^(binomial(n,2))*q_pochhammer(n-1, -1/4, 1/4) for n in (1..15)] # G. C. Greubel, Feb 21 2021

G.f. A(x) satisfies: A(x) = 1 + x * A(4*x) / (1 - x).
G.f.: Sum_{k>=0} 2^(k*(k - 1)) * x^k / Product_{j=0..k-1} (1 - 4^j*x).
a(0) = 1; a(n) = Sum_{k=0..n-1} 4^k * a(k).
a(n) ~ c * 2^(n*(n - 1)), where c = Product_{k>=1} (1 + 1/4^k) = 1.355909673863479380345544...
a(n) = 4^(binomial(n+1,2))*(-1/4; 1/4){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021

cp's OEIS Frontend

A285101 a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)), where A242378(n,a(n-1)) shifts the prime factorization of a(n-1) n primes towards larger primes with A003961.

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A155103 Triangle read by rows: Matrix inverse of A155102.

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A269455 Number of Type I (singly-even) self-dual binary codes of length 2n.

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A322299 Number of distinct automorphism group sizes for binary self-dual codes of length 2n.

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A006088 a(n) = (2^n + 2) a(n-1) (kissing number of Barnes-Wall lattice in dimension 2^n).

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A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

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A290000 a(n) = Product_{k=1..n-1} (3^k + 1).

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