A227681 -id:A227681 - cp's OEIS frontend

A152398 The q-exponential of x, e_q(x,q), evaluated at q = -x.

1, 1, 1, 2, 4, 7, 11, 17, 28, 48, 80, 128, 204, 332, 545, 887, 1432, 2313, 3750, 6086, 9859, 15944, 25788, 41749, 67604, 109415, 177017, 286409, 463495, 750081, 1213713, 1963771, 3177444, 5141446, 8319390, 13461189, 21780519, 35241682

Offset: 0

Paul D. Hanna, Dec 16 2008

nonn

The g.f.s for this sequence illustrate the following formula:
log(e_q(x,q)) = Sum_{n>=1} (1-q)^n/(1-q^n)*x^n/n, where
e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential of x and
faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1) is the q-factorial of n.

			G.f.: e_q(x,-x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 11*x^6 + ...
log(e_q(x,-x)) = x + x^2/2 + 4*x^3/3 + 9*x^4/4 + 16*x^5/5 + 22*x^6/6 + ... (A152399).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein, q-Exponential Function from MathWorld.
Eric Weisstein, q-Factorial from MathWorld.

Cf. A152399: log(e_q(x, -x)); A227681, A306749.

a(n)=polcoeff(sum(k=0,n,x^k/(prod(j=1,k,(1-(-x)^j)/(1+x))+x*O(x^n))),n)

a(n)=polcoeff(exp(sum(k=1,n,x^k*(1+x)^k/(1-(-x)^k)/k)+x*O(x^n)),n)

{a(n)=polcoeff(1/prod(k=1,n,1+(1+x)*(-x)^k+x*O(x^n)),n)} \\ Paul D. Hanna, Dec 20 2008

G.f.: e_q(x,-x) = Sum_{n>=0} x^n/(Product_{k=1..n} (1-(-x)^k)/(1+x)).
G.f.: e_q(x,-x) = exp( Sum_{n>=1} x^n*(1+x)^n/(1-(-x)^n)/n ).
G.f.: 1/Product_{k>0} 1+(1+x)*(-x)^k. - Vladeta Jovovic, Dec 19 2008
a(n) ~ c/r^n where r = (sqrt(5) - 1)/2 = 0.6180339887... and c = 0.652419554233497352459208493304650..., where e_q(-r,r) = 0.887276226980250304353751667447441... - Paul D. Hanna, Dec 20 2008
c = 1 / (r * sqrt(5) * QPochhammer((1-sqrt(5))/2)). - Vaclav Kotesovec, Oct 22 2020

A160571 G.f.: Product_{n>=1} (1 + x^n + x^(n+1)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 28, 38, 52, 70, 92, 119, 154, 200, 258, 329, 416, 523, 655, 819, 1022, 1269, 1566, 1924, 2357, 2879, 3507, 4263, 5170, 6250, 7530, 9048, 10849, 12980, 15496, 18466, 21967, 26079, 30894, 36526, 43109, 50792, 59743, 70160

Offset: 0

Paul D. Hanna, May 20 2009, May 21 2009, Jul 17 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 10*x^6 + 15*x^7 + ...
G.f.: A(x) = (1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))*(1+x^4*(1+x))*...
G.f.: A(x) = (1+x*(1+x)) + x^2*(1+x)*(1 + x^3*(1+x))*(1+x*(1+x))/(1-x) + x^7*(1+x)^2*(1 + x^5*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))/((1-x)*(1-x^2)) + x^15*(1+x)^3*(1 + x^7*(1+x))*(1+x*(1+x))*(1+x^2*(1+x))*(1+x^3*(1+x))/((1-x)*(1-x^2)*(1-x^3)) + ...
G.f.: A(x) = 1 + x*(1+x)/(1-x) + x^3*(1+x)^2/((1-x)*(1-x^2)) + x^6*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*(1+x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...

Robert Israel, Table of n, a(n) for n = 0..10000
MathOverflow, Being even or odd in the product expansion, 2017.

Cf. A151552, A227681.

N:= 100: # for a(0)..a(N)
P:= mul(1+x^n+x^(n+1),n=1..N):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Sep 04 2018

With[{nn=50},CoefficientList[Series[Product[1+x^n+x^(n+1),{n,1,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Dec 29 2015 *)

a(n)=polcoeff(prod(k=1,n,1+x^k*(1+x) +x*O(x^n)),n)

{a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(3*m+1)/2)*(1+x)^m*(1 + x^(2*m+1)*A)*prod(k=1, m, (1+A*x^k)/(1-x^k+x*O(x^n)))); polcoeff(A, n)}

{a(n)=local(A=1+x); A=sum(m=0, n, x^(m*(m+1)/2)*(1+x)^m/prod(k=1, m, 1-x^k +x*O(x^n))); polcoeff(A, n)}

G.f.: A(x) = Sum_{n>=0} x^(n*(3*n+1)/2)*(1+x)^n*(1 + x^(2*n+1)*(1+x)) * Product_{k=1..n} (1 + x^k*(1+x))/(1-x^k) due to Sylvester's identity.
G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2)*(1+x)^n / Product_{k=1..n} (1-x^k).
G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (-1)^(d+1)*(1 + x)^d/d). - Ilya Gutkovskiy, Apr 18 2019
a(n) ~ c * exp(r*sqrt(n)) / n^(3/4), where r = 2*sqrt(-polylog(2,-2)) = 2.397287105779... and c = (-polylog(2,-2))^(1/4) / (6*sqrt(Pi)) = 0.10294821957... - Vaclav Kotesovec, Oct 24 2020, updated Jun 25 2021

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

A276987 Decimal expansion of (phi-1)_inf = (1/phi)_inf, where (q)_inf is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

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

Offset: 0

Vladimir Reshetnikov, Sep 24 2016

(1/phi)_inf appears as a coefficient in asymptotics of A274983, A274985.

			0.1208019218617061294237231569887920563...

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol, Golden Ratio.

Cf. A274983, A274985, A062073, A227681.

RealDigits[QPochhammer[1/GoldenRatio], 10, 100][[1]]

(1/phi)inf = Product{k > 0} (1 - 1/phi^k).

A306565 Expansion of Product_{k>=1} (1 - x^k * (1 + x)).

Original entry on oeis.org

1, -1, -2, -1, 1, 3, 4, 3, 1, -2, -6, -8, -8, -8, -5, 2, 8, 12, 17, 22, 23, 17, 7, 0, -7, -22, -40, -51, -53, -49, -45, -42, -30, -4, 30, 65, 90, 100, 112, 137, 157, 152, 120, 71, 18, -33, -80, -125, -187, -275, -357, -401, -407, -380, -327, -269, -221, -171, -75, 102, 322, 515, 669, 801

Offset: 0

Seiichi Manyama, Apr 16 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Convolution inverse of A227681.

Cf. A160571.

m = 63; CoefficientList[Series[Product[1 - x^k * (1 + x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k*(1+x)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, (1+x)^d/d))))

G.f.: exp( - Sum_{k>=1} x^k * Sum_{d|k} (1+x)^d / d).

A336975 Expansion of Product_{k>=1} 1/(1 - x^k * (k + x)).

Original entry on oeis.org

1, 1, 4, 9, 22, 47, 107, 221, 468, 953, 1932, 3814, 7560, 14625, 28192, 53757, 101827, 190907, 356362, 659716, 1215314, 2224968, 4053914, 7346367, 13260001, 23822114, 42629786, 75991017, 134991954, 238948942, 421656911, 741750026, 1301116634, 2275985891, 3971022904

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006906, A227681, A336976, A336977, A336978, A336979, A336980.

m = 34; CoefficientList[Series[Product[1/(1 - x^k*(k + x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k*(k+x)))

N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (k/d+x)^d/d))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (k/d + x)^d / d).

a(n) ~ c * n * phi^(n+1) / 5, where c = Product_{k>=3} 1/(1 - 1/phi^k*(k + 1/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 06 2021

A336976 Expansion of Product_{k>=1} 1/(1 - x^k * (1 + k*x)).

Original entry on oeis.org

1, 1, 3, 7, 15, 32, 65, 131, 260, 501, 965, 1825, 3419, 6326, 11652, 21230, 38405, 69015, 123334, 218980, 386809, 679757, 1189360, 2071761, 3594325, 6211826, 10698409, 18363038, 31420994, 53605525, 91198970, 154746133, 261929303, 442310873, 745264674, 1253081340, 2102754561

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A227681, A336975, A336977, A336978, A336979, A336980.

m = 36; CoefficientList[Series[Product[1/(1 - x^k*(1 + k*x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k*(1+k*x)))

N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (1+k/d*x)^d/d))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (1 + k/d * x)^d / d).

a(n) ~ c * phi^(n+1) / sqrt(5), where c = Product_{k>=2} 1/(1 - 1/phi^k*(1 + k/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 06 2021

A306749 Expansion of Product_{k>=1} 1/(1 - x^k * (1 - x)).

Original entry on oeis.org

1, 1, 1, 0, 0, -1, 1, -1, 2, -2, 2, -4, 6, -8, 11, -13, 16, -23, 32, -44, 61, -80, 102, -133, 178, -243, 331, -441, 579, -759, 1001, -1335, 1792, -2398, 3186, -4205, 5537, -7320, 9734, -12975, 17266, -22893, 30267, -40004, 52968, -70282, 93348, -123900, 164179, -217277

Offset: 0

Seiichi Manyama, Apr 16 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A152398, A227681, A306691.

m = 49; CoefficientList[Series[Product[1/(1 - x^k * (1 - x)), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, 1-x^k*(1-x)))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (1 - x)^d/d). - Ilya Gutkovskiy, Apr 16 2019

A307676 Expansion of Product_{k>=1} (1 - x^k(1 - x))/(1 - x^k(1 + x)).

Original entry on oeis.org

1, 0, 2, 4, 6, 14, 22, 46, 74, 138, 236, 406, 698, 1182, 1994, 3342, 5590, 9274, 15386, 25380, 41818, 68670, 112586, 184210, 300940, 490962, 800026, 1302278, 2118008, 3442042, 5590092, 9073632, 14720738, 23872776, 38700910, 62720726, 101622398, 164617032

Offset: 0

Seiichi Manyama, Apr 21 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A227681, A198197.

nmax = 50; CoefficientList[Series[Product[(1 - x^k*(1 - x))/(1 - x^k*(1 + x)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k*(1-x))/(1-x^k*(1+x))))

N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, ((1+x)^d-(1-x)^d)/d))))

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} ((1+x)^d - (1-x)^d)/d).

a(n) ~ phi^(n+4) / sqrt(5), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 31 2021

A309172 Expansion of Product_{k>=1} 1/(1 - (1 + x + x^2) * x^k).

Original entry on oeis.org

1, 1, 3, 7, 15, 31, 64, 128, 254, 496, 961, 1844, 3516, 6662, 12564, 23593, 44153, 82385, 153351, 284857, 528235, 978148, 1809120, 3342722, 6171318, 11385733, 20994298, 38693809, 71288111, 131297855, 241761727, 445068646, 819205061, 1507641487, 2774307387, 5104712633

Offset: 0

Ilya Gutkovskiy, Jul 15 2019

nonn

Cf. A160571, A227681, A309173.

nmax = 35; CoefficientList[Series[Product[1/(1 - (1 + x + x^2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Exp[Sum[x^k Sum[(1 + x + x^2)^d/d, {d, Divisors[k]}], {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^k * Sum_{d|k} (1 + x + x^2)^d/d).

a(n) ~ 1/((1 + 2*r + 3*r^2) * QPochhammer[r] * r^(n+1)), where r = A192918. - Vaclav Kotesovec, Jul 16 2019

cp's OEIS Frontend

A152398 The q-exponential of x, e_q(x,q), evaluated at q = -x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A160571 G.f.: Product_{n>=1} (1 + x^n + x^(n+1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A276987 Decimal expansion of (phi-1)_inf = (1/phi)_inf, where (q)_inf is the q-Pochhammer symbol, phi = (1+sqrt(5))/2 is the golden ratio.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A306565 Expansion of Product_{k>=1} (1 - x^k * (1 + x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A336975 Expansion of Product_{k>=1} 1/(1 - x^k * (k + x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A336976 Expansion of Product_{k>=1} 1/(1 - x^k * (1 + k*x)).

Views

Author

Keywords

Links

Crossrefs

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

A307676 Expansion of Product_{k>=1} (1 - x^k(1 - x))/(1 - x^k(1 + x)).