A160571 -id:A160571 - 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

A336980 Expansion of Product_{k>=1} (1 + x^k * (1 + k*x)).

Original entry on oeis.org

1, 1, 2, 4, 8, 13, 22, 39, 65, 104, 160, 263, 413, 646, 975, 1479, 2198, 3354, 5017, 7389, 10770, 15721, 22668, 32663, 47200, 67761, 96389, 135977, 191431, 268805, 376211, 523692, 730301, 1014029, 1401553, 1925074, 2638522, 3608182, 4924194, 6694070, 9088239, 12323707, 16668255

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

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

Cf. A160571, A336975, A336976, A336977, A336978, A336979.

m = 42; CoefficientList[Series[Product[1 + x^k*(1 + k*x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Apr 29 2021 *)

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

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

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).

A336979 Expansion of Product_{k>=1} (1 + x^k * (k + x)).

Original entry on oeis.org

1, 1, 3, 6, 11, 21, 37, 69, 108, 192, 312, 522, 827, 1297, 2032, 3240, 4982, 7569, 11508, 17107, 25696, 38340, 57080, 83298, 121373, 175653, 253455, 364307, 523650, 747487, 1063375, 1498471, 2106317, 2955154, 4124071, 5750547, 8000706, 11104596, 15324290, 21093106

Offset: 0

Seiichi Manyama, Aug 09 2020

nonn

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

Cf. A022629, A160571, A336975, A336976, A336977, A336978, A336980.

m = 39; CoefficientList[Series[Product[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(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, (-1)^(d+1)*(k/d+x)^d/d))))

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

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

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

A309173 Expansion of Product_{k>=1} (1 + (1 + x + x^2) * x^k).

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 15, 23, 34, 50, 71, 100, 140, 195, 268, 363, 487, 650, 865, 1145, 1505, 1962, 2541, 3275, 4208, 5390, 6879, 8740, 11053, 13917, 17459, 21837, 27244, 33906, 42085, 52085, 64268, 79071, 97025, 118772, 145082, 176869, 215204, 261333, 316705, 383019

Offset: 0

Ilya Gutkovskiy, Jul 15 2019

nonn

Cf. A160571, A227681, A309172.

nmax = 45; CoefficientList[Series[Product[(1 + (1 + x + x^2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[x^k Sum[(-1)^(d + 1) (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)^(d+1) * (1 + x + x^2)^d/d).

A345729 Expansion of Product_{k>=1} (1 + x^k + x^(k+2)).

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 7, 11, 15, 20, 28, 36, 50, 62, 86, 105, 141, 175, 226, 283, 358, 446, 557, 691, 852, 1055, 1286, 1587, 1918, 2353, 2830, 3445, 4134, 4993, 5977, 7174, 8555, 10220, 12138, 14436, 17092, 20232, 23896, 28158, 33172, 38937, 45736, 53512, 62662

Offset: 0

Vaclav Kotesovec, Jun 25 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A160571.

nmax = 60; CoefficientList[Series[Product[1 + x^k + x^(k+2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * exp(r*sqrt(n)) / n^(3/4), where r = 2*sqrt(-polylog(2,-2)) and c = (-polylog(2,-2))^(1/4) / (6*sqrt(3*Pi)).

A346679 Expansion of Product_{k>=1} (1 + x^k(1+x)) / (1 - x^k(1+x)).

Original entry on oeis.org

1, 2, 6, 14, 32, 68, 140, 280, 544, 1034, 1930, 3544, 6416, 11472, 20288, 35532, 61696, 106304, 181906, 309362, 523228, 880576, 1475424, 2462302, 4094682, 6787588, 11219504, 18498094, 30429502, 49955706, 81864400, 133940690, 218834842, 357090226, 582050680

Offset: 0

Vaclav Kotesovec, Jul 31 2021

nonn

Convolution of A160571 and A227681.

Cf. A160571, A227681.

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

a(n) ~ 2*c * phi^(n+1) / sqrt(5), where phi = A001622 is the golden ratio and c = Product_{k>=2} (phi^k + 1 + 1/phi) / (phi^k - 1 - 1/phi) = 32.9911047431709572178149423384235021321790640826498395008790713974339...

cp's OEIS Frontend

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

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A336980 Expansion of Product_{k>=1} (1 + x^k * (1 + k*x)).

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

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A336979 Expansion of Product_{k>=1} (1 + x^k * (k + x)).

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

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

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A309173 Expansion of Product_{k>=1} (1 + (1 + x + x^2) * x^k).

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A345729 Expansion of Product_{k>=1} (1 + x^k + x^(k+2)).

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A346679 Expansion of Product_{k>=1} (1 + x^k(1+x)) / (1 - x^k(1+x)).