A048651 -id:A048651 - cp's OEIS frontend

A100221 Decimal expansion of Product_{k>=1} (1-1/4^k).

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

Offset: 0

Eric W. Weisstein, Nov 09 2004

			0.68853753712033971545651435729350818467554981937833...

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.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A027637, A048651, A100220, A100222.

Cf. A000203, A132020, A132034, A132035, A132036, A132037, A132038, A258459.

EllipticThetaPrime[1, 0, 1/2]^(1/3)/2^(1/4)
N[QPochhammer[1/4]] (* G. C. Greubel, Nov 30 2015 *)
RealDigits[Fold[Times,1-1/4^Range[1000]],10,110][[1]] (* Harvey P. Dale, Sep 27 2024 *)

prodinf(x=1, 1-1/4^x) \\ Altug Alkan, Dec 01 2015

Equals exp(-Sum_{k>0} sigma_1(k)/(k*4^k)) where sigma_1() is A000203(). - Hieronymus Fischer, Aug 07 2007
Equals (1/4; 1/4){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(Pi/log(2)) * exp(log(2)/12 - Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027637(n). (End)

A238351 Number of compositions p(1)+p(2)+...+p(k) = n such that for no part p(i) = i (compositions without fixed points).

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 11, 22, 42, 82, 161, 316, 624, 1235, 2449, 4864, 9676, 19267, 38399, 76582, 152819, 305085, 609282, 1217140, 2431992, 4860306, 9714696, 19419870, 38824406, 77624110, 155208405, 310352615, 620601689, 1241036325, 2481803050, 4963170896

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

nonn

Column k=0 of A238349 and of A238350.

			The a(7) = 22 such compositions are:
01:  [ 2 1 1 1 1 1 ]
02:  [ 2 1 1 1 2 ]
03:  [ 2 1 1 2 1 ]
04:  [ 2 1 1 3 ]
05:  [ 2 1 2 1 1 ]
06:  [ 2 1 2 2 ]
07:  [ 2 1 4 ]
08:  [ 2 3 1 1 ]
09:  [ 2 3 2 ]
10:  [ 2 4 1 ]
11:  [ 2 5 ]
12:  [ 3 1 1 1 1 ]
13:  [ 3 1 1 2 ]
14:  [ 3 1 2 1 ]
15:  [ 3 3 1 ]
16:  [ 3 4 ]
17:  [ 4 1 1 1 ]
18:  [ 4 1 2 ]
19:  [ 4 3 ]
20:  [ 5 1 1 ]
21:  [ 6 1 ]
22:  [ 7 ]

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 = 0..1000
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

b:= proc(n, i) option remember; `if`(n=0, 1,
       add(`if`(i=j, 0, b(n-j, i+1)), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, b[n-j, i+1]], {j, 1, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2014, after Maple *)

a(n) ~ c * 2^n, where c = A048651/2 = 0.14439404754330121... - Vaclav Kotesovec, May 01 2014

A259149 Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.99812906992595851327996232224527387813073843536581646175407814028299858...

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)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292821, A292825, A292829.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-2Pi]], 10, 104] // 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(-2*Pi)) = exp(Pi/12)*Gamma(1/4)/(2*Pi^(3/4)).

A259148 Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.954918789987674103751233978110291077632715373807805283148799191676094...

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)), A259147 phi(exp(-Pi/2)), 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)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292820, A292824, A292828.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi]], 10, 104] // 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)) = exp(Pi/24)*Gamma(1/4)/(2^(7/8)*Pi^(3/4)).
Equals 1/exp(A255695). - Hugo Pfoertner, May 28 2025

A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.99999651264548223429509891685211924765750978932634584844773269100472...

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), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A000706, A292822, A292826.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-4*Pi]], 10, 103] // 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(-4*Pi)) = exp(Pi/6)*Gamma(1/4)/(2^(11/8)*Pi^(3/4)).
A259150 = A259148 * exp(Pi/8)/sqrt(2). - Vaclav Kotesovec, Jul 03 2017

A065446 Decimal expansion of Product_{k>=1} (1-1/2^k)^(-1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 18 2001

			3.46274661945506361153795734292443116454075790290...

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

Harry J. Smith, Table of n, a(n) for n=1..2000
Steven R. Finch, Digital Search Tree Constants [Broken link]
Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]
Igor Pak, Greta Panova, and Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018. [p. 13]
Jonathan Sondow and Eric W. Weisstein, MathWorld: Wallis Formula.

Cf. A000041, A002884, A048651, A048652, A006099.

evalf(1+sum(2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # Robert FERREOL, Feb 22 2020

N[ Product[ 1/(1 - 1/2^k), {k, 1, Infinity} ], 500 ]
RealDigits[1/QPochhammer[1/2, 1/2], 10, 100][[1]] (* Vaclav Kotesovec, Jun 22 2014 *)

{ default(realprecision, 2080); x=prodinf(k=1, 1/(1 - 1/2^k)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065446.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009

prodinf(k=1, 1/(1-1/2^k)) \\ Michel Marcus, Feb 22 2020

Equals Sum_{n>=0} 1/A002884(n)*Product_{j=1..n} 2^j/(2^j-1). - Geoffrey Critzer, Jun 30 2017
Equals 1/QPochhammer(1/2, 1/2){infinity}. - _G. C. Greubel, Jan 18 2018
Equals 1 + Sum_{n>=1} 2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). - Robert FERREOL, Feb 22 2020
Equals 1 / A048651 (constant). - Hugo Pfoertner, Nov 28 2020
Equals Sum_{n>=0} A000041(n)/2^n. - Amiram Eldar, Jan 19 2021

More terms from Robert G. Wilson v, Nov 19 2001

Further terms from Vladeta Jovovic, Dec 01 2001

A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 216, 228, 240, 294, 308, 322, 384, 400, 416, 729, 756, 783, 900, 930, 960, 1089, 1122, 1155, 1728, 1776, 1824, 2028, 2080, 2132, 2352, 2408, 2464, 3375, 3450, 3525, 3840, 3920, 4000, 4335

Offset: 1

Hieronymus Fischer, Aug 13 2007, Aug 20 2007

nonn

If n is written in base 3 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 d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m= -2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(11)=floor(11/3^0)*floor(11/3^1)*floor(11/3^2)=11*3*1=33;
a(13)=52 since 13=111(base-3) and so a(13)=111*11*1(base-3)=13*4*1=52.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A048651, A098844, A067080, A132019, A100220, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132028(p=4)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Table[(f = If[# < 3, #, #*f[Quotient[#, 3]]] &)[n], {n, 51}] (* Ivan Neretin, May 29 2016 *)

Recurrence: a(n)=n*a(floor(n/3)); a(n*3^m)=n^m*3^(m(m+1)/2)*a(n).
a(k*3^m)=k^(m+1)*3^(m(m+1)/2), for k=1 or 2.
a(n)<=b(n), where b(n)=n^(1+floor(log_3(n)))/3^(1/2*(1+floor(log_3(n)))*floor(log_3(n))); equality holds if n is a power of 3 or two times a power of 3.
Also: a(n)<=2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=1.1364507...*3^A000217(log_3(n)), equality for n=2*3^m, m>=0.
a(n)>c*b(n), where c=0.3826631966790330232889550... (see constant A132019).
Also: a(n)>c*2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=0.434877...*3^A000217(log_3(n)).
lim inf a(n)/b(n)=0.3826631966790330232889550..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_3(n))/2)=0.3826631966790330232889550...*sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_3(n))/2)=sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
lim inf a(n)/a(n+1)=0.3826631966790330232889550... for n-->oo (see constant A132019).
a(n)=O(n^((1+log_3(n))/2)).

A132034 Decimal expansion of Product_{k>0} (1-1/6^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.805687728162164940923750...

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. A000203, A027873, A048651, A067080, A098844, A100220, A132022, A132026, A132038.

digits = 103; NProduct[1-1/6^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+20] // N[#, digits+20]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/6,1/6]] (* G. C. Greubel, Nov 30 2015 *)

prodinf(x=1, 1-(1/6)^x) \\ Altug Alkan, Dec 01 2015

Equals exp( -Sum_{n>0} sigma_1(n)/(n*6^n) ).
Equals (1/6; 1/6){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(6)) * exp(log(6)/24 - Pi^2/(6*log(6))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(6))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027873(n). (End)

A177510 Number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i and pi >= p0.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 25, 46, 87, 167, 324, 634, 1248, 2466, 4887, 9706, 19308, 38455, 76659, 152925, 305232, 609488, 1217429, 2432399, 4860881, 9715511, 19421029, 38826059, 77626471, 155211785, 310357462, 620608652, 1241046343, 2481817484, 4963191718, 9925669171, 19850186856, 39698516655, 79394037319

Offset: 0

Mats Granvik, Dec 11 2010

nonn

a(0)=1, otherwise row sums of A179748.
For n>=1 cumulative sums of A008930.
a(n) is proportional to A048651*A000079. The error (a(n)-A048651*A000079) divided by sequence A186425 tends to the golden ratio A001622. This can be seen when using about 1000 decimals of the constant A048651 = 0.2887880950866024212... - [Mats Granvik, Jan 01 2015]
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer compositions of n with exactly one part on or above the diagonal. For example, the a(1) = 1 through a(5) = 8 compositions are:
  (1)  (2)   (3)    (4)     (5)
       (11)  (21)   (31)    (41)
             (111)  (112)   (212)
                    (211)   (311)
                    (1111)  (1112)
                            (1121)
                            (2111)
                            (11111)
(End)

			From _Joerg Arndt_, Mar 24 2014: (Start)
The a(7) = 25 such compositions are:
01:  [ 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 3 ]
05:  [ 1 1 1 2 1 1 ]
06:  [ 1 1 1 2 2 ]
07:  [ 1 1 1 3 1 ]
08:  [ 1 1 1 4 ]
09:  [ 1 1 2 1 1 1 ]
10:  [ 1 1 2 1 2 ]
11:  [ 1 1 2 2 1 ]
12:  [ 1 1 2 3 ]
13:  [ 1 1 3 1 1 ]
14:  [ 1 1 3 2 ]
15:  [ 1 2 1 1 1 1 ]
16:  [ 1 2 1 1 2 ]
17:  [ 1 2 1 2 1 ]
18:  [ 1 2 1 3 ]
19:  [ 1 2 2 1 1 ]
20:  [ 1 2 2 2 ]
21:  [ 1 2 3 1 ]
22:  [ 2 2 3 ]
23:  [ 2 3 2 ]
24:  [ 3 4 ]
25:  [ 7 ]
(End)

Cf. A238859 (compositions with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
The version for partitions is A001477, strong A002620.
The version for permutations is A057427, strong A000295.
The opposite version is A238874, first column of A352522.
The version for fixed points is A240736, nonfixed A352520.
The strong version is A351983, column k=1 of A352524.
This is column k = 1 of A352525.
A238349 counts compositions by fixed points, first col A238351.
A352517 counts weak excedances of standard compositions.
Cf. A008930, A010054, A088218, A098825, A114088, A219282, A238352, A319005, A350839, A352488, A352489, A352514, A352515, A352516.

A179748 := proc(n,k) option remember; if k= 1 then 1; elif k> n then 0 ; else add( procname(n-i,k-1),i=1..k-1) ; end if; end proc:
A177510 := proc(n) add(A179748(n,k),k=1..n) ;end proc:
seq(A177510(n),n=1..20) ; # R. J. Mathar, Dec 14 2010

Clear[t, nn]; nn = 39; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *)
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pdw[#]==1&]],{n,0,10}] (* Gus Wiseman, Mar 31 2022 *)

N=66; q='q+O('q^N); Vec( 1 + q/(1-q) * sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014

@CachedFunction
def T(n, k): # A179748
    if n == 0:  return int(k==0);
    if k == 1:  return int(n>=1);
    return sum( T(n-i, k-1) for i in [1..k-1] );
# to display triangle A179748 including column zero = [1,0,0,0,...]:
#for n in [0..10]: print([ T(n,k) for k in [0..n] ])
def a(n): return sum( T(n,k) for k in [0..n] )
print([a(n) for n in [0..66]])
# Joerg Arndt, Mar 24 2014

G.f.: 1 + q/(1-q) * sum(n>=0, q^n * prod(k=1..n, (1-q^k)/(1-q) ) ). [Joerg Arndt, Mar 24 2014]

New name and a(0) = 1 prepended, Joerg Arndt, Mar 24 2014

A259151 Decimal expansion of phi(exp(-8*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.999999999987838443290442788140998270959486945673852198543872725583699...

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), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), 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)), A363117 phi(exp(-7*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-8*Pi]], 10, 103] // 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(-8*Pi)) = (sqrt(2) - 1)^(1/4)*exp(Pi/3)*(Gamma(1/4)/(2^(29/16)*Pi^(3/4))).
A259151 = A259147 * exp(5*Pi/16)/2. - Vaclav Kotesovec, Jul 03 2017

cp's OEIS Frontend

A100221 Decimal expansion of Product_{k>=1} (1-1/4^k).

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A238351 Number of compositions p(1)+p(2)+...+p(k) = n such that for no part p(i) = i (compositions without fixed points).

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A259149 Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A259148 Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A065446 Decimal expansion of Product_{k>=1} (1-1/2^k)^(-1).

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A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.

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