A048651 -id:A048651 - cp's OEIS frontend

A132269 a(n) = Product_{k>=0} (1 + floor(n/2^k)).

1, 2, 6, 8, 30, 36, 56, 64, 270, 300, 396, 432, 728, 784, 960, 1024, 4590, 4860, 5700, 6000, 8316, 8712, 9936, 10368, 18200, 18928, 21168, 21952, 27840, 28800, 31744, 32768, 151470, 156060, 170100, 174960, 210900, 216600, 234000, 240000, 340956, 349272, 374616

Offset: 0

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base 2 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product (1+d(m)d(m-1)d(m-2)...d(2)d(1)d(0))*(1+d(m)d(m-1)d(m-2)...d(2)d(1))*(1+d(m)d(m-1)d(m-2)...d(2))*...*(1+d(m)d(m-1)d(m-2))*(1+d(m)d(m-1))*(1+d(m)).
From Gary W. Adamson, Aug 25 2016: (Start)
Given the following production matrix M =
  1, 0, 0, 0, 0, ...
  2, 0, 0, 0, 0, ...
  0, 3, 0, 0, 0, ...
  0, 4, 0, 0, 0, ...
  0, 0, 5, 0, 0, ...
  0, 0, 6, 0, 0, ...
  0, 0, 0, 7, 0, ...
  ...
the sequence is the left-shifted vector as lim_{n->infinity} M^n. (End)

			a(10) = (1 + floor(10/2^0))*(1 + floor(10/2^1))*(1 + floor(10/2^2))*(1 + floor(10/2^3)) = 11*6*3*2 = 396;
a(17) = 4860 since 17 = 10001_2 and so a(17) = (1+10001_2)*(1+1000_2)*(1+100_2)*(1+10_2)*(1+1) = 18*9*5*3*2 = 4860.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A048651, A081845, A132270, A132271(p=10), A132272, A132327(p=3), A132328.
For formulas regarding a general parameter p (i.e., terms 1+floor(n/p^k)) see A132271.
For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.

[1] cat [n le 1 select 2 else (1+n)*Self(Floor(n/2)): n in [1..50]]; // Vincenzo Librandi, Aug 26 2016

f:= proc(n) option remember; (1+n)*procname(floor(n/2)) end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Aug 26 2016

Table[Product[1 + Floor[2 n/2^k], {k, 2 n}], {n, 0, 42}] (* or *)
Table[Function[w, Times @@ Map[1 + FromDigits[PadRight[w, #], 2] &, Range@ Length@ w]]@ IntegerDigits[n, 2], {n, 0, 42}] (* Michael De Vlieger, Aug 26 2016 *)

Recurrence: a(n)=(1+n)*a(floor(n/2)); a(2n)=(1+2n)*a(n); a(n*2^m) = (Product_{k=1..m} (1 + n*2^k))*a(n).
a(2^m-1) = 2^(m*(m+1)/2), a(2^m) = 2^(m*(m+1)/2)*Product_{k=0..m} (1 + 1/2^k), m>=1.
a(n) = A132270(2n) = (1+n)*A132270(n).
Asymptotic behavior: a(n) = O(n^((1+log_2(n))/2)); this follows from the inequalities below.
a(n) <= A098844(n)*Product_{k=0..floor(log_2(n))} (1 + 1/2^k).
a(n) >= A098844(n)/Product_{k=1..floor(log_2(n))} (1 - 1/2^k).
a(n) < c*n^((1+log_2(n))/2) = c*2^A000217(log_2(n)), where c = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627... (see constant A081845).
a(n) > n^((1+log_2(n))/2) = 2^A000217(log_2(n)),
lim sup a(n)/A098844(n) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627..., for n->oo (see constant A081845).
lim inf a(n)/A098844(n) = 1/Product_{k>=1} (1 - 1/2^k) = 1/0.288788095086602421..., for n->oo (see constant A048651).
lim inf a(n)/n^((1+log_2(n))/2) = 1, for n->oo.
lim sup a(n)/n^((1+log_2(n))/2) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627..., for n->oo (see constant A081845).
lim inf a(n+1)/a(n) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627... for n->oo (see constant A081845).
G.f. g(x) satisfies g(x) = (1+2x)*g(x^2) + 2*x^2*(1+x)*g'(x^2). - Robert Israel, Aug 26 2016

A132035 Decimal expansion of Product_{k>0} (1-1/7^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.8367954070890378710...

Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A027875, A048651, A100220, A132023, A132026, A132038, A098844, A067080, A000203.

digits = 103; NProduct[1-1/7^k, {k, 1, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/7], 10, 100][[1]] (* Amiram Eldar, May 09 2023 *)

prodinf(k=1, 1 - 1/(7^k)) \\ Amiram Eldar, May 09 2023

Equals exp(-Sum_{n>0} sigma_1(n)/(n*7^n)) = exp(-Sum_{n>0} A000203(n)/(n*7^n)).
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(7)) * exp(log(7)/24 - Pi^2/(6*log(7))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(7))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027875(n). (End)

A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.74931147780000278742962565878338031190409252790117392831206731...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A259151 phi(exp(-8*Pi)), A363019 phi(exp(-10*Pi)), A363020 phi(exp(-12*Pi)), A292864 phi(exp(-16*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292819, A292823, A292827.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi/2]], 10, 105] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-Pi/2)) = ((sqrt(2) - 1)^(1/3)*(4 + 3*sqrt(2))^(1/24) * exp(Pi/48) * Gamma(1/4))/(2^(5/6)*Pi^(3/4)).
phi(exp(-Pi/2)) = (sqrt(2)-1)^(1/4) * exp(Pi/48) * Gamma(1/4)/(2^(13/16)*Pi^(3/4)). - Vaclav Kotesovec, Jul 03 2017

A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).

Original entry on oeis.org

1, 4, 12, 36, 92, 228, 540, 1236, 2748, 6004, 12876, 27252, 57036, 118308, 243564, 498564, 1015484, 2060484, 4167804, 8409588, 16934748, 34049940, 68378220, 137185428, 275026476, 551052676, 1103618508, 2209525092, 4422484764, 8850120420, 17707920924

Offset: 0

Vaclav Kotesovec, Aug 25 2015

nonn

Cf. A032302, A070933, A261563.

nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1)), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 - QPochhammer[-2, x]/(3 QPochhammer[2, x]))[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = c * 2^n, where c = 1/(A048651 * A083864) = 2*Product_{j>=1} (2^j+1)/(2^j-1) = 16.5119758715565001310882816988645462530540032335764606912075051272567456...

A075271 a(0) = 1 and, for n >= 1, (BM)a(n) = 2*a(n-1), where BM is the BinomialMean transform.

Original entry on oeis.org

1, 3, 17, 211, 5793, 339491, 41326513, 10282961907, 5181436229441, 5258784071302723, 10717167529963833681, 43779339268428732008723, 358114286723184561034838497, 5862685570087914880854259126371, 192026370558313054275618817346778353

Offset: 0

John W. Layman, Sep 11 2002

The BinomialMean transform BM is defined by (BM)a(n) = (M^n)a(0) where (M)a(n) is the mean (a(n) + a(n+1))/2, or, alternatively, by (BM)a(n) = (Sum_{k=0..n} binomial(n,k)*a(k))/(2^n).

The BinomialMean transform of this sequence is given in A075272.

			Given that a(0)=1 and a(1)=3. Then (BM)a(2) = (1 + 2*3 + a(2))/4 = 2a(1) = 6, hence a(2)=17.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A075272, A048651.

iBM:= proc(p) proc(n) option remember; add(2^(k)*p(k)*(-1)^(n-k) *binomial(n, k), k=0..n) end end: a:= iBM(aa): aa:= n-> `if`(n=0, 1, 2*a(n-1)): seq(a(n), n=0..16);  # Alois P. Heinz, Sep 09 2008

iBM[p_] := Module[{proc}, proc[n_] := proc[n] = Sum[2^k*p[k]*(-1)^(n-k) * Binomial[n, k], {k, 0, n}]; proc]; a = iBM[aa]; aa[n_] := If[n == 0, 1, 2*a[n-1]]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *)
Table[Sum[QFactorial[k, 2] Binomial[n + 1, k]/2, {k, 0, n + 1}], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 16 2016 *)

O.g.f. as a continued fraction: A(x) = 1/(1 + x - 2^2*x/(1 - 2*(2 - 1)^2*x/(1 + x - 2^4*x/(1 - 2*(2^2 - 1)^2*x/(1 + x - 2^6*x/(1 - 2*(2^3 - 1)^2*x/(1 + x - 2^8*x/(1 - 2*(2^4 - 1)^2*x/(1 + x - ... ))))))))). Cf. A075272. - Peter Bala, Nov 10 2017

a(n) ~ A048651 * 2^(n*(n+3)/2). - Vaclav Kotesovec, Jun 09 2025

More terms from Alois P. Heinz, Sep 09 2008

A132268 Decimal expansion of Product_{k>0} (1-1/12^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 20 2007

			0.9097262689059948886363620469770...

G. C. Greubel, Table of n, a(n) for n = 0..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A027880, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132267, A132323-A132326, A000203.

digits = 106; NProduct[1-1/12^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/12,1/12]] (* G. C. Greubel, Dec 05 2015 *)

prodinf(k=1, (1-1/12^k)) \\ Michel Marcus, Dec 05 2015

Equals exp(-Sum_{n>0} sigma_1(n)/(n*12^n)) = exp(-Sum_{n>0} A000203(n)/(n*12^n)).
Equals (1/12; 1/12){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 05 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(12)) * exp(log(12)/24 - Pi^2/(6*log(12))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(12))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027880(n). (End)

A240736 Number of compositions of n having exactly one fixed point.

Original entry on oeis.org

1, 1, 1, 4, 7, 16, 29, 60, 120, 238, 479, 956, 1910, 3817, 7633, 15252, 30491, 60955, 121865, 243650, 487165, 974112, 1947851, 3895086, 7789153, 15576624, 31150481, 62296424, 124585395, 249158607, 498297297, 996562085, 1993071152, 3986055928, 7971971230

Offset: 1

Joerg Arndt and Alois P. Heinz, Apr 11 2014

nonn

			a(4) = 4: 13, 22, 112, 1111.
a(5) = 7: 14, 32, 131, 221, 1112, 1121, 11111.

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Column k=1 of A238349 and of A238350.

b:= proc(n, i) option remember; `if`(n=0, 1, series(
      add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n), x, 2))
    end:
a:= n-> coeff(b(n, 1), x, 1):
seq(a(n), n=1..40);

b[n_, i_] := b[n, i] = If[n == 0, 1, Series[Sum[b[n - j, i + 1]*If[i == j, x, 1], {j, 1, n}], {x, 0, 2}]]; a[n_] := SeriesCoefficient[b[n, 1], {x, 0, 1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 06 2014, after Maple *)

a(n) ~ c * 2^n, where c = A065442 * A048651 / 2 = 0.2319972162254452238942023675457837005318389885... - Vaclav Kotesovec, Sep 06 2014

A227682 G.f.: exp( Sum_{n>=1} x^n / (n(1-x)^n (1-x^n)) ).

Original entry on oeis.org

1, 1, 3, 7, 16, 35, 76, 162, 342, 715, 1484, 3060, 6278, 12824, 26102, 52969, 107224, 216601, 436798, 879584, 1769117, 3554726, 7136736, 14318524, 28711315, 57544864, 115290624, 230910993, 462362571, 925610398, 1852669016, 3707705019, 7419275371, 14844857959

Offset: 0

Paul D. Hanna, Jul 19 2013

nonn

Number of compositions of n with k sorts of parts k where the sorts of parts are nondecreasing through the composition, see example. - Joerg Arndt, May 01 2014

			From _Joerg Arndt_, May 01 2014: (Start)
The a(5) = 35 compositions as described in the first comment are (here p:s stands for a part p of sort s)
01:  [ 1:0  1:0  1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:0  2:0  ]
03:  [ 1:0  1:0  1:0  2:1  ]
04:  [ 1:0  1:0  2:0  1:0  ]
05:  [ 1:0  1:0  3:0  ]
06:  [ 1:0  1:0  3:1  ]
07:  [ 1:0  1:0  3:2  ]
08:  [ 1:0  2:0  1:0  1:0  ]
09:  [ 1:0  2:0  2:0  ]
10:  [ 1:0  2:0  2:1  ]
11:  [ 1:0  2:1  2:1  ]
12:  [ 1:0  3:0  1:0  ]
13:  [ 1:0  4:0  ]
14:  [ 1:0  4:1  ]
15:  [ 1:0  4:2  ]
16:  [ 1:0  4:3  ]
17:  [ 2:0  1:0  1:0  1:0  ]
18:  [ 2:0  1:0  2:0  ]
19:  [ 2:0  1:0  2:1  ]
20:  [ 2:0  2:0  1:0  ]
21:  [ 2:0  3:0  ]
22:  [ 2:0  3:1  ]
23:  [ 2:0  3:2  ]
24:  [ 2:1  3:1  ]
25:  [ 2:1  3:2  ]
26:  [ 3:0  1:0  1:0  ]
27:  [ 3:0  2:0  ]
28:  [ 3:0  2:1  ]
29:  [ 3:1  2:1  ]
30:  [ 4:0  1:0  ]
31:  [ 5:0  ]
32:  [ 5:1  ]
33:  [ 5:2  ]
34:  [ 5:3  ]
35:  [ 5:4  ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A227681, A238349, A253829.

Flatten[{1,Table[SeriesCoefficient[Exp[Sum[x^k / (k*(1-x)^k * (1-x^k)),{k,1,n}]],{x,0,n}], {n,1,40}]}] (* Vaclav Kotesovec, May 01 2014 *)

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(m*(1-x)^m*(1-x^m +x*O(x^n))) )), n)}
for(n=0, 50, print1(a(n), ", "))

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m*sumdiv(m, d, 1/(1-x +x*O(x^n))^d/d) )), n)}
for(n=0, 50, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} x^n * Sum_{d|n} 1/(d*(1-x)^d) ).
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 35*x^5 + 76*x^6 + 162*x^7 +...
where
log(A(x)) = x/((1-x)*(1-x)) + x^2/(2*(1-x)^2*(1-x^2)) + x^3/(3*(1-x)^3*(1-x^3)) + x^4/(4*(1-x)^4*(1-x^4)) + x^5/(5*(1-x)^5*(1-x^5)) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 56*x^5/5 + 107*x^6/6 + 197*x^7/7 + 365*x^8/8 + 679*x^9/9 + 1280*x^10/10 +...
a(n) = A238350(n*(n+3)/2,n), a(n) is the number of compositions of n*(n+3)/2 with exactly n fixed points. - Alois P. Heinz, Apr 11 2014
a(n) ~ c * 2^n, where c = 1/(2*A048651) = 1.73137330972753180576... - Vaclav Kotesovec, May 01 2014
G.f.: Product {n >= 1} 1/(1 - x^n/(1 - x)). Row sums of A253829. - Peter Bala, Jan 20 2015

A322211 a(n) = coefficient of x^n*y^n in Product_{n>=1} 1/(1 - (x^n + y^n)).

Original entry on oeis.org

1, 2, 10, 38, 158, 602, 2382, 9142, 35492, 136936, 530404, 2053848, 7972272, 30977742, 120576112, 469915012, 1833813534, 7164469910, 28021000340, 109699469798, 429850240742, 1685728936622, 6615913739206, 25983523253950, 102115250446680, 401557335718522, 1579978592844064, 6219928993470190, 24498287876663618, 96535916978924934, 380568644820360668

Offset: 0

Paul D. Hanna, Nov 30 2018

nonn

Number of subsets of partitions of 2n that have sum n. Olivier Gérard, May 07 2020

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 38*x^3 + 158*x^4 + 602*x^5 + 2382*x^6 + 9142*x^7 + 35492*x^8 + 136936*x^9 + 530404*x^10 + 2053848*x^11 + 7972272*x^12 + ...
RELATED SERIES.
The product P(x,y) = Product_{n>=1} 1/(1 - (x^n + y^n)) begins
P(x,y) = 1 + (x + y) + (2*x^2 + 2*x*y + 2*y^2) + (3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3) + (5*x^4 + 7*x^3*y + 10*x^2*y^2 + 7*x*y^3 + 5*y^4) + (7*x^5 + 12*x^4*y + 18*x^3*y^2 + 18*x^2*y^3 + 12*x*y^4 + 7*y^5) + (11*x^6 + 19*x^5*y + 34*x^4*y^2 + 38*x^3*y^3 + 34*x^2*y^4 + 19*x*y^5 + 11*y^6) + (15*x^7 + 30*x^6*y + 56*x^5*y^2 + 74*x^4*y^3 + 74*x^3*y^4 + 56*x^2*y^5 + 30*x*y^6 + 15*y^7) + (22*x^8 + 45*x^7*y + 94*x^6*y^2 + 133*x^5*y^3 + 158*x^4*y^4 + 133*x^3*y^5 + 94*x^2*y^6 + 45*x*y^7 + 22*y^8) + ...
in which this sequence equals the coefficients of x^n*y^n for n >= 0.
The logarithm of the g.f. begins
log( A(x) ) = 2*x + 16*x^2/2 + 62*x^3/3 + 272*x^4/4 + 922*x^5/5 + 3640*x^6/6 + 12966*x^7/7 + 49872*x^8/8 + 190340*x^9/9 + 745316*x^10/10 + 2928136*x^11/11 + 11602184*x^12/12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..500 (previous b-file of terms 0..50 supplied by Vaclav Kotesovec).

Cf. A108796, A258471, A322210, A322213, A322198.

nmax = 20; s = Series[Product[1/(1 - (x^k + y^k)), {k, 1, nmax}], {x, 0, nmax}, {y, 0, nmax}]; Flatten[{1, Table[Coefficient[s, x^n*y^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Dec 04 2018 *)

{P = 1/prod(n=1,61, (1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{a(n) = polcoeff( polcoeff( P,n,x),n,y)}
for(n=0,35, print1( a(n),", ") )

Main diagonal of square table A322210.

a(n) ~ c * 4^n / sqrt(Pi*n), where c = 1 / A048651 = 1 / Product_{k>=1} (1 - 1/2^k) = 3.46274661945506361153795734292443116454075790290443839... - Vaclav Kotesovec, Dec 23 2018

A132020 Decimal expansion of Product_{k>=0} (1 - 1/(2*4^k)).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

This is the limiting probability that a large random symmetric binary matrix is nonsingular (cf. A086812, A048651). In other words, equals Lim_{n->oo} A086812(n)/A006125(n+1).- H. Tracy Hall, Sep 07 2024

			0.41942244179510759770995610770297425223395323439266674908044991663177...

John E. Cremona and R. W. K. Odoni, Some density results for negative Pell equations; an application of graph theory, Journal of the London Mathematical Society s2-39:1 (1989), pp. 16-28.

Cf. A004171, A048651, A098844, A067080, A132019, A132026, A132028, A100221, A000217, A081845.

evalf(1+sum((-1)^n*2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # Robert FERREOL, Feb 23 2020

RealDigits[ Product[1 - 1/(2*4^i), {i, 0, 175}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
RealDigits[QPochhammer[1/2, 1/4], 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)

prodinf(k=0,1-1.>>(2*k+1)) \\ Charles R Greathouse IV, Nov 16 2012

Equals lim inf_{n->oo} Product_{k=0..floor(log_4(n))} floor(n/4^k)*4^k/n.
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^((1/2)*(1+floor(log_4(n)))*floor(log_4(n))).
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^A000217(floor(log_4(n))).
Equals (1/2)*exp(-Sum_{n>0} (4^(-n)*(Sum_{k|n} 1/(k*2^k)))).
Equals lim inf_{n->oo} A132028(n)/A132028(n+1).
Equals Product_{k>0} (1-1/(2^k+1)). - Robert G. Wilson v, May 25 2011
From Robert FERREOL, Feb 23 2020: (Start)
Equals Product_{k>0} (1 + 1/2^k)^(-1) = 2/A081845.
Equals 1 + Sum_{n>=1} (-1)^n*2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). (End)
From Peter Bala, Jan 15 2021: (Start)
Constant C = Sum_{n >= 0} 2^n/Product_{k = 1..n} (1 - 4^k).
Faster converging series:
2*C = (1/2)*Sum_{n >= 0} 2^(-n)/Product_{k = 1..n} (1 - 4^k);
(2^4)*C = 7*Sum_{n >= 0} 2^(-3*n)/Product_{k = 1..n} (1 - 4^k);
(2^9)*C = 7*31*Sum_{n >= 0} 2^(-5*n)/Product_{k = 1..n} (1 - 4^k), and so on.
Slower converging series:
C = -Sum_{n >= 0} 2^(3*n)/Product_{k = 1..n} (1 - 4^k);
7*C = Sum_{n >= 0} 2^(5*n)/Product_{k = 1..n} (1 - 4^k);
7*31*C = -Sum_{n >= 0} 2^(7*n)/Product_{k = 1..n} (1 - 4^k), and so on. (End)
Equals Product_{n>=0} (1 - 1/A004171(n)). - Amiram Eldar, May 09 2023

Name corrected by Charles R Greathouse IV, Nov 16 2012

cp's OEIS Frontend

A132269 a(n) = Product_{k>=0} (1 + floor(n/2^k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A132035 Decimal expansion of Product_{k>0} (1-1/7^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A261584 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - 2*x^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A075271 a(0) = 1 and, for n >= 1, (BM)a(n) = 2*a(n-1), where BM is the BinomialMean transform.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A132268 Decimal expansion of Product_{k>0} (1-1/12^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A240736 Number of compositions of n having exactly one fixed point.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A227682 G.f.: exp( Sum_{n>=1} x^n / (n*(1-x)^n * (1-x^n)) ).

A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).

A227682 G.f.: exp( Sum_{n>=1} x^n / (n(1-x)^n (1-x^n)) ).