A005329 -id:A005329 - cp's OEIS frontend

A027880 a(n) = Product_{i=1..n} (12^i - 1).

1, 11, 1573, 2716571, 56328099685, 14016177372718235, 41852067359921313500005, 1499635200191700040518673659035, 644815685260091508353787979063721364325, 3327107302821620489265827570792988872583047378075

Offset: 0

N. J. A. Sloane

nonn

In general, Product_{i=1..n} (q^i-1) ~ c * q^(n*(n+1)/2), where c = Product_{k >= 1} (1-1/q^k). - Vaclav Kotesovec, Nov 21 2015

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), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11).

Cf. A132268.

[1] cat [&*[12^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

FoldList[Times,1,12^Range[10]-1] (* Harvey P. Dale, Mar 01 2015 *)
Abs@QPochhammer[12, 12, Range[0, 30]] (* G. C. Greubel, Nov 24 2015 *)

a(n) = prod(k=1, n, 12^k - 1) \\ Altug Alkan, Nov 25 2015

a(n) ~ c * 12^(n*(n+1)/2), where c = Product_{k>=1} (1-1/12^k) = 0.909726268905994888636362046977080249120791691941... . - Vaclav Kotesovec, Nov 21 2015
(11)^n*(13)^(floor(n/2))|a(n) for n>=0. - G. C. Greubel, Nov 24 2015
Equals 12^(binomial(n+1,2))*(1/12;1/12){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 12^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 12^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132268. - Amiram Eldar, May 07 2023

A028365 Order of general affine group over GF(2), AGL(n,2).

Original entry on oeis.org

1, 2, 24, 1344, 322560, 319979520, 1290157424640, 20972799094947840, 1369104324918194995200, 358201502736997192984166400, 375234700595146883504949480652800, 1573079924978208093254925489963584716800

Offset: 0

N. J. A. Sloane

For n > 0, a(n) = v(n+1)/v(n), where v = A203305 is the Vandermonde determinant of the first n of the numbers -2^j - 1; see the Mathematica section. - Clark Kimberling, Jan 01 2012

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 54 (1.64).

Seiichi Manyama, Table of n, a(n) for n = 0..57
Abdalla G. M. Ahmed, Matt Pharr, Victor Ostromoukhov, and Hui Huang, SZ Sequences: Binary-Based (0, 2^q)-Sequences, arXiv:2505.20434 [cs.GR], 2025. See p. 7.
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
Putnam Competition 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.

Cf. A000217, A002884, A005329, A020522, A048651, A053601, A203305.

[1] cat [(&*[2^(n+1) - 2^(j+1): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 31 2023

A028365 := n->2^n*product(2^n-2^'i','i'=0..n-1); # version 1
A028365 := n->product(2^'j'-1,'j'=1..n)*2^binomial(n+1,2); # version 2

RecurrenceTable[{a[1]==1, a[2]==2, a[3]==24, a[n]==(6a[n-1]^2 a[n-3] - 8a[n-1] a[n-2]^2)/(a[n-2] a[n-3])}, a[n], {n,20}] (* Harvey P. Dale, Aug 03 2011 *)
(* Next, the connection with Vandermonde determinants *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]   (* A203303 *)
Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *) (* Clark Kimberling, Jan 01 2011 *)
Table[(-1)^n*2^Binomial[n+1,2]*QPochhammer[2,2,n], {n,0,20}] (* G. C. Greubel, Aug 31 2023 *)

a(n)=if(n<0,0,prod(k=1,n,2^k-1)*2^((n^2+n)/2)) /* Michael Somos, May 09 2005 */

[product(2^(n+1) - 2^(k+1) for k in range(n)) for n in range(21)] # G. C. Greubel, Aug 31 2023

a(n) is asymptotic to C*2^(n*(n+1)) where C = 0.288788095086602421278899721... = prod(k>=1, 1-1/2^k) (cf. A048651). - Benoit Cloitre, Apr 11 2003
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)). [From Putman Exam]. - Max Alekseyev, May 18 2007
a(n) = 2*A203305(n), n > 0. - Clark Kimberling, Jan 01 2012
From Max Alekseyev, Jun 09 2015: (Start)
a(n) = 2^A000217(n) * A005329(n).
a(n) = 2^n * A002884(n).
a(n) = 2^n * n! * A053601(n). (End)
From G. C. Greubel, Aug 31 2023: (Start)
a(n) = Product_{j=0..n-1} (2^(n+1) - 2^(j+1)).
a(n) = (-1)^n * 2^binomial(n+1,2) * QPochhammer(2,2,n). (End)

A289545 Number of flags in an n-dimensional vector space over GF(2).

Original entry on oeis.org

1, 1, 4, 36, 696, 27808, 2257888, 369572160, 121459776768, 79991977040128, 105466641591287296, 278244130564826548224, 1468496684404408240109568, 15502543140842029367582248960, 327332729703063815298568073396224, 13823536566775628445052117519260598272

Offset: 0

Geoffrey Critzer, Jul 28 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..80
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A381299.

Column k=2 of A381426.

b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
      b(u+o, 0$2), 0)+add(2^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..16);  # Alois P. Heinz, Feb 21 2025

nn = 15; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,
   nn}] CoefficientList[Series[ 1/(1 - (eq[z] - 1)) /. q -> 2, {z, 0, nn}], z]

a(n) = Sum A005329(n)/( A005329(n_1)*A005329(n_2)*...*A005329(n_k) ) where the sum is over all compositions of n = n_1 + n_2 + ... + n_k.
G.f. a(n)/A005329(n) is the coefficient of x^n in 1/(2 - eq(x)) where eq(x) is the 2-exponential function.
a(n) = Sum_{k=0..binomial(n,2)} 2^k * A381299(n,k). - Alois P. Heinz, Feb 21 2025

A135950 Matrix inverse of triangle A022166.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -8, 14, -7, 1, 64, -120, 70, -15, 1, -1024, 1984, -1240, 310, -31, 1, 32768, -64512, 41664, -11160, 1302, -63, 1, -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1, 268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1

Offset: 0

Paul D. Hanna, Dec 08 2007

A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.
The coefficient [x^k] of Product_{i=1..n} (x-2^(i-1)). - Roger L. Bagula, Mar 20 2009
Triangle T(n,k), 0 <= k <= n, read by rows given by (-1, 1-q, -q^2, q-q^3, -q^4, q^2-q^5, -q^6, q^3-q^7, -q^8, ...) DELTA (1, 0, q, 0, q^2, 0, q^3, 0, q^4, 0, ...) (for q = 2) = (-1, -1, -4, -6, -16, -28, -64, -120, -256, ...) DELTA (1, 0, 2, 0, 4, 0, 8, 0, 16, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2013
Reversed rows of triangle A158474. - Werner Schulte, Apr 06 2019
T(n,k) = Sum mu(0,U) where the sum is taken over the subspaces U of GF(2)^n having dimension n-k and mu is the Moebius function of the poset of all subspaces of GF(2)^n. - Geoffrey Critzer, Jun 02 2024

			Triangle begins:
         1;
        -1,       1;
         2,      -3,        1;
        -8,      14,       -7,      1;
        64,    -120,       70,    -15,      1;
     -1024,    1984,    -1240,    310,    -31,    1;
     32768,  -64512,    41664, -11160,   1302,  -63,    1;
  -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1; ...

G. C. Greubel, Rows n = 0..75 of triangle, flattened

Cf. A022166, A006125, A028361, A127850, A135951 (central terms), A158474.

max = 9; M = Table[QBinomial[n, k, 2], {n, 0, max}, {k, 0, max}] // Inverse; Table[M[[n, k]], {n, 1, max+1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)
p[x_, n_, q_] := (-1)^n*q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
Table[CoefficientList[Series[p[x, n, 2], {x, 0, n}], x], {n, 0, 10}]// Flatten (* G. C. Greubel, Apr 15 2019 *)

T(n,k)=local(q=2,A=matrix(n+1,n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1);A[n+1,k+1]

Unsigned column 0 equals A006125(n) = 2^(n*(n-1)/2).
Unsigned column 1 equals A127850(n) = (2^n-1)*2^(n*(n-1)/2)/2^(n-1).
Row sums equal 0^n.
Unsigned row sums equal A028361(n) = Product_{k=0..n} (1+2^k).
T(n,k) = (-1)^(n-k) * A022166(n,k) * 2^binomial(n-k,2) for 0 <= k <= n. - Werner Schulte, Apr 06 2019 [corrected by Werner Schulte, Dec 27 2021]
Sum_{n>=0} Sum_{k=0..n} T(n,k)y^k*x^n/A005329(n) = e(y*x)/e(x) where e(x) = Sum_{n>=0} x^n/A005329(n). - Geoffrey Critzer, Jun 02 2024

A377484 a(n) = Product_{d|n, d>1} (d - 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 10, 6, 21, 16, 36, 10, 330, 12, 78, 112, 315, 16, 1360, 18, 2052, 240, 210, 22, 53130, 96, 300, 416, 6318, 28, 146160, 30, 9765, 640, 528, 816, 1570800, 36, 666, 912, 560196, 40, 639600, 42, 27090, 39424, 990, 46, 37456650, 288, 42336, 1600, 45900, 52, 1874080, 2160

Offset: 1

Ridouane Oudra, Oct 29 2024

			a(12) = (2-1)*(3-1)*(4-1)*(6-1)*(12-1) = 1*2*3*5*11 = 330.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007955, A020696.

Cf. A000079, A005329, A027871, A027872, A027875, A027879, A175787.

with(numtheory): seq(mul(d-1, d in divisors(n) minus {1}), n=1..80);

a[n_] := Times @@ (Rest@ Divisors[n] - 1); Array[a, 60] (* Amiram Eldar, Nov 01 2024 *)

a(n) = my(d=divisors(n)); prod(k=2, #d, d[k]-1); \\ Michel Marcus, Oct 30 2024

a(n) = Product_{k=2..A000005(n)} (A027750(n,k) - 1).
a(p^n) = Product_{k=1..n} (p^k - 1), where p is prime, and n an integer.
a(2^n) = A005329(n).
a(3^n) = A027871(n).
a(5^n) = A027872(n).
a(7^n) = A027875(n).
a(11^n) = A027879(n).
From Amiram Eldar, Nov 02 2024: (Start)
a(n) = n-1 if and only if n is in A175787 (i.e., n = 4 or n is prime).
a(n) == 1 (mod 2) if and only if n is a power of 2 (A000079). (End)

A139382 Triangle read by rows, T(n,k) = (2^k-1) * T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 11, 1, 1, 40, 90, 26, 1, 1, 121, 670, 480, 57, 1, 1, 364, 4811, 7870, 2247, 120, 1, 1, 1093, 34041, 122861, 77527, 9807, 247, 1, 1, 3280, 239380, 1876956, 2526198, 695368, 41176, 502, 1, 1, 9841, 1678940, 28393720, 80189094, 46334382, 5924720, 169186, 1013, 1

Offset: 1

Gary W. Adamson, Apr 16 2008

Row sums = A135922 starting with offset 1: (1, 2, 6, 26, 158, 1330, ...).
This triangle is the q-analog of A008277 (Stirling numbers of the 2nd kind) for q=2 (see Cai et al. link). - Werner Schulte, Apr 04 2019
T(n,k) is the number of naturally labeled posets on [n] with height at most one containing exactly k minimal elements.  See link by David Bevan and others below. - Geoffrey Critzer, May 03 2025

			First few rows of the triangle are:
  1;
  1,   1;
  1,   4,   1;
  1,  13,  11,   1;
  1,  40,  90,  26,   1;
  1, 121, 670, 480,  57,   1;
  ...
a(13) = T(5,3) = 90 = (2^3 - 1)*T(4,3) + T(4,2) = 7*11 + 13.

G. C. Greubel, Rows n = 1..100 of triangle, flattened
David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
Yue Cai, Richard Ehrenborg, and Margaret A. Readdy, q-Stirling numbers revisited, arXiv:1706.06623 [math.CO], 2017.
Yue Cai, Richard Ehrenborg, and Margaret A. Readdy, q-Stirling numbers revisited, Electronic Journal of Combinatorics, 25 Issue 1 (2018), Paper #P1.37.

Cf. A008277, A135922, A333143.

Cf. A000295 (2nd diagonal), A003462 (column 2), A016212 (column 3), A156823.

# Uses[qStirling2 from A333143]
seq(seq(qStirling2(n, k, 2), k=0..n), n=0..9); # Peter Luschny, Mar 10 2020
# Alternative.
A139382 := proc(n, k) if k = 1 then 1 elif k = n then 1 elif k < 1 then 0 else
(2^k - 1)*A139382(n-1, k) + A139382(n-1, k-1) fi end:
for n from 1 to 8 do seq(A139382(n, k), k = 1..n) od; # Peter Luschny, Jun 28 2022

T[1, 1]:= 1; T[n_, k_]:= T[n, k] = If[k > n || k < 1, 0, (2^k-1)*T[n-1, k] + T[n-1, k-1]]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] (* G. C. Greubel, Apr 02 2019 *)

{T(n,k) = if(k<1 || k>n, 0, if(n==1 && k==1, 1, (2^k-1)*T(n-1,k) + T(n-1,k-1)))};
for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 02 2019

@CachedFunction
def T(n, k):
   if (k==1): return 1
   elif (k==n): return 1
   else: return (2^k-1)*T(n-1, k) + T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Apr 02 2019

Triangle read by rows, T(n,k) = (2^k-1) * T(n-1,k) + T(n-1,k-1). Let X = an infinite bidiagonal matrix with (1,3,7,15,31...) in the main diagonal and (1,1,1,...) in the subdiagonal. n-th row of the triangle = X^n * [1,0,0,0,...].
From Werner Schulte, Apr 02 2019: (Start)
G.f. of column k: col(k,t) = Sum_{n>=k} T(n,k)*t^n = t^k/Product_{i=1..k} (1 - (2^i-1)*t) for k > 0.
Sum_{k>0} col(k,t) * (Product_{i=1..k-1} (1 - 2^i)) = t (empty product equals 1).
Sum_{k=1..n} (-1)^k * 2^binomial(k,2) * T(n,k) = (-1)^n for n > 0.
An example for k=3: g.f. of column 3: col(3,t) = Sum_{n>=3} T(n,3) * t^n = 1*t^3 + 11*t^4 + 90*t^5 + 670*t^6 + ... = t^3 * (1 + 11*t + 90*t^2 + 670*t^3 + ...) = t^3 / Product_{i=1..3} (1 - (2^i - 1)*t) = t^3 / ((1 - t) * (1 - 3*t) * (1 - 7*t)) = t^3 / (1 - 11*t + 31*t^2 - 21*t^3). Perhaps the following recurrence formula is useful too: col(k,t) = col(k-1,t) * t / (1 - (2^k - 1)*t) for k > 1 with initial value col(1,t) = t / (1 - t). Finally: col(k,t) is the g.f. of column k.
With regard to the 2nd formula: We can it replace with the following formula: Sum_{k=1..n} T(n,k) * (Product_{i=1..k-1} (1-2^i)) = A000007(n-1) for n > 0 with empty product 1 (case k=1). Example for n=5: 1*1 + (-1)*40 + (-1)*(-3)*90 + (-1)*(-3)*(-7)*26 + (-1)*(-3)*(-7)*(-15)*1 = 0. (End)
T(n,k) = (1/(2^binomial(k,2)*A005329(k))) * Sum_{j=0..k} (-1)^(k-j)*2^binomial(k-j,2)*A022166(k,j)*(2^j-1)^n. - Fabian Pereyra, Jan 27 2024
T(n,k) = Sum_{j=k..n} (-1)^(n-j)*binomial(n,j)*qBinomial(j,k,2), where qBinomial(n,k,2) is A022166(n,k). - Fabian Pereyra, Jan 31 2024

More terms from G. C. Greubel, Apr 02 2019

A203305 Vandermonde determinant of the first n terms of (1,3,7,15,31,...).

Original entry on oeis.org

1, 2, 48, 64512, 20808990720, 6658450862270054400, 8590449816558320728896700416000, 180165778137909187135292776823951671626301440000, 246665746050863452218796304775365273357060390005370386894553088000000

Offset: 1

Clark Kimberling, Jan 01 2012

nonn

Each term divides its successor, as in A028365 and A203307.

G. C. Greubel, Table of n, a(n) for n = 1..21

Cf. A000225, A005329, A028365, A203307, A335011.

[1] cat [(&*[(&*[2^(k+1) - 2^j: j in [1..k]]): k in [1..n-1]]): n in [2..15]]; // G. C. Greubel, Aug 30 2023

(* First program *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]         (* A203305 *)
Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
%/2                         (* A203307 *)
(* Second program *)
Table[(-1)^n * 2^(n*(n+1)*(2*n+1)/6 - 1) / QPochhammer[2, 2, n] * Product[QPochhammer[1/2^k, 2, k], {k, 2, n}], {n, 10}] (* Vaclav Kotesovec, Feb 18 2021 *)

[product(product(2^k - 2^j for j in range(1,k)) for k in range(2,n+1)) for n in range(1,16)] # G. C. Greubel, Aug 30 2023

a(n) = Product_{k=1..n-1} Product_{j=1..k} (2^(k+1) - 2^j).
From Vaclav Kotesovec, Feb 18 2021: (Start)
a(n) = (-1)^n * (2^(n*(n+1)*(2*n+1)/6 - 1) / QPochhammer(2,2,n)) * Product_{k=2..n} QPochhammer(1/2^k, 2, k).
a(n) ~ 2^(n*(n^2 - 1)/3) * QPochhammer(1/2)^n / A335011. (End)
a(n) = Product_{k=2..n} ( 2^(k+1)^2 * QPochhammer(2^(-k-1), 2, k+1) )/ (2^(k+1) - 1). - G. C. Greubel, Aug 30 2023

A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

Original entry on oeis.org

1, 0, 1, 6, 91, 2820, 177661, 22562946, 5753551231, 2940064679040, 3007686166657921, 6156733583148764286, 25211824022994189751171, 206510050572345408251841660, 3383254158526734823389921915781

Offset: 1

N. J. A. Sloane

nonn

q-Subfactorial for q=2. - Vladimir Reshetnikov, Sep 12 2016

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.
T. L. Greenough and T. Lockman, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..50
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]
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.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Subfactorial, q-Factorial, q-Analog.

Cf. A002820, A005329, A005321.

p:=proc(n) if n=0 then 1 else product(2^i-1,i=1..n) fi end: a:=n->p(n-1)*sum((-1)^j/p(j),j=0..n-1): seq(a(n),n=1..17); # Emeric Deutsch, Jan 23 2005

a[1] = 1; a[n_] := a[n] = (2^(n-1)-1)*a[n-1] + (-1)^(n-1); Array[a, 15] (* Jean-François Alcover, Apr 05 2016, after Max Alekseyev *)
With[{q = 2}, Table[QFactorial[n, q] Sum[(-1)^k/QFactorial[k, q], {k, 0, n}], {n, 0, 20}]] (* Vladimir Reshetnikov, Sep 12 2016 *)

For n>1, a(n) = (2^(n-1)-1)*a(n-1) + (-1)^(n-1). - Max Alekseyev, Feb 23 2012
a(n) = p(n-1)*sum((-1)^j/p(j), j=0..n-1), where p(0) = 1, p(k) = product(2^i-1, i=1..k) for k>0. - Emeric Deutsch, Jan 23 2005
a(n) ~ A048651^2 * 2^(n*(n-1)/2). - Vaclav Kotesovec, Oct 09 2019

More terms from Max Alekseyev, Apr 27 2010

A182507 G.f.: Sum_{n>=0} n! * 2^(n(n-1)/2) x^n / Product_{k=1..n} (1 + k2^kx).

Original entry on oeis.org

1, 1, 2, 12, 232, 12848, 1858464, 663242944, 562426769024, 1103780804371200, 4916976475489286656, 48986367134323580374016, 1078808700869188981508990976, 52024935094126934151475827453952, 5451309776848243787358722272838524928

Offset: 0

Paul D. Hanna, May 03 2012

nonn

Compare the g.f. to the identities:
(1) 1/(1-x) = Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + k*x).
(2) 1+x = Sum_{n>=0} 2^(n*(n-1)/2) * x^n / Product_{k=1..n} (1 + 2^k*x).
First differs from A309615 at a(5) = 12848, A309615(5) = 19230. - Gus Wiseman, Aug 11 2019

			G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 232*x^4 + 12848*x^5 + 1858464*x^6 +...
such that
A(x) = 1 + x/(1+2*x) + 2!*2^1*x^2/((1+1*2*x)*(1+2*4*x)) + 3!*2^3*x^3/((1+1*2*x)*(1+2*4*x)*(1+3*8*x)) + 4!*2^6*x^4/((1+1*2*x)*(1+2*4*x)*(1+3*8*x)*(1+4*16*x)) +...

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. A005329, A182489, A309615.

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

A048652 Continued fraction for Product_{k >= 1} (1-1/2^k) (Cf. A048651).

Original entry on oeis.org

0, 3, 2, 6, 4, 1, 2, 1, 9, 2, 1, 2, 3, 2, 3, 5, 1, 2, 1, 1, 6, 1, 2, 5, 79, 6, 4, 5, 1, 1, 1, 1, 12, 1, 1, 2, 5, 1, 659, 2, 17, 1, 5, 2, 3, 2, 6, 1, 1, 2, 3, 1, 2, 6, 1, 1, 3, 11, 1, 1, 2, 1, 1, 2, 4, 11, 2, 1, 3, 4, 2, 2, 1, 3, 1, 71, 1, 1, 1, 19, 1, 4, 1, 1, 8, 1, 49, 3, 1, 2, 2, 11, 1, 11, 10, 1, 2, 1, 1

Offset: 0

N. J. A. Sloane

Continued fraction expansion of the constant Product{k>=1} (1-1/2^k)^(-1) = 3.46274661945506361... (A065446) gives essentially the same sequence.

			0.2887880950866024212788997219294585937270...
0.288788095086602421278899721... = 0 + 1/(3 + 1/(2 + 1/(6 + 1/(4 + ...)))). - _Harry J. Smith_, May 02 2009

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Steven R. Finch, Minkowski's Question Mark Function [Broken link]
Steven R. Finch, Minkowski's Question Mark Function [From the Wayback machine]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A005329, A048651, A065446.

ContinuedFraction[ N[ Product[ 1/(1 - 1/2^k), {k, 1, Infinity} ], 500 ], 49]

{ allocatemem(932245000); default(realprecision, 21000); x=prodinf(k=1, -1/2^k, 1); z=contfrac(x); for (n=1, 20001, write("b048652.txt", n-1, " ", z[n])); } \\ Harry J. Smith, May 07 2009

Corrected by Harry J. Smith, May 02 2009

Deleted old PARI program. - Harry J. Smith, May 20 2009

cp's OEIS Frontend

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A182507 G.f.: Sum_{n>=0} n! * 2^(n(n-1)/2) x^n / Product_{k=1..n} (1 + k2^kx).