A306749 -id:A306749 - cp's OEIS frontend

A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).

1, 1, 3, 6, 12, 23, 43, 79, 142, 252, 442, 766, 1316, 2244, 3799, 6393, 10704, 17841, 29618, 49000, 80823, 132964, 218242, 357501, 584608, 954553, 1556575, 2535425, 4125805, 6708143, 10898897, 17696749, 28719276, 46586050, 75538702, 122444483, 198420445, 321461918

Offset: 0

Paul D. Hanna, Jul 19 2013

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 23*x^5 + 43*x^6 + 79*x^7 +...
where
log(A(x)) = x*(1+x)/(1-x) + x^2*(1+x)^2/(2*(1-x^2)) + x^3*(1+x)^3/(3*(1-x^3)) + x^4*(1+x)^4/(4*(1-x^4)) + x^5*(1+x)^5/(5*(1-x^5)) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 10*x^3/3 + 17*x^4/4 + 26*x^5/5 + 38*x^6/6 + 57*x^7/7 + 81*x^8/8 + 118*x^9/9 + 180*x^10/10 +...

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

Cf. A160571, A227682, A306565, A306749.

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

{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+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+x)^d / d ).
G.f.: Product {n >= 1} 1/(1 - (1 + x)*x^n). - Peter Bala, Jan 20 2015
a(n) ~ phi^(n+1) / (sqrt(5)* A276987), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 16 2019

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

Original entry on oeis.org

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

A306691 Expansion of Product_{k>=1} (1 + x^k * (1 - x)).

Original entry on oeis.org

1, 1, 0, 1, -1, 1, 0, -1, 1, 0, 0, -2, 4, -4, 3, -2, 0, 2, -1, -2, 3, -1, -3, 8, -11, 10, -8, 9, -13, 15, -9, -2, 6, 2, -14, 21, -20, 10, 8, -25, 29, -22, 20, -31, 46, -53, 48, -33, 15, -9, 27, -57, 71, -62, 53, -63, 83, -87, 53, 16, -80, 91, -47, 3, -19, 99, -186, 219, -184

Offset: 0

Seiichi Manyama, Apr 16 2019

sign

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

Cf. A160571, A198197, A306565, A306749.

With[{nn=70},CoefficientList[Series[Product[1+x^k (1-x),{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Aug 03 2020 *)

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

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

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A227681 G.f.: exp( Sum_{n>=1} x^n * (1+x)^n / (n*(1-x^n)) ).

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

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A306691 Expansion of Product_{k>=1} (1 + x^k * (1 - x)).

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