A299106 -id:A299106 - cp's OEIS frontend

A302017 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k-1))).

1, 1, 2, 3, 6, 11, 21, 39, 73, 137, 257, 482, 903, 1693, 3173, 5948, 11149, 20899, 39174, 73430, 137641, 258002, 483614, 906513, 1699219, 3185111, 5970352, 11191163, 20977346, 39321116, 73705711, 138158128, 258971363, 485430483, 909918190, 1705601814, 3197075934, 5992778881, 11233201667

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

nonn

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Antidiagonal sums of absolute values of A286352.

Cf. A000700, A299106, A299208.

nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[1/(1 - x QPochhammer[x^2]^2/(QPochhammer[x] QPochhammer[x^4])), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + (-x)^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A000700(k-1)*a(n-k).
a(n) ~ c / r^n, where r = 0.5334880525001986092393688937248506539793821912... is the root of the equation 1 + r - r^2 * QPochhammer(-1/r, r^2) = 0 and c = 0.48000092330632206397886602198643227268597451507794232644772186731542555975... = (2*(1 + r)*Log[r])/(2*(2 + r)*Log[r] + (1 + r)*Log[1 - r^2] + (1 + r) * QPolyGamma[Log[-1/r] / Log[r^2], r^2] + 4*r^4*Log[r] * Derivative[0,1][QPochhammer][-1/r, r^2]). - Vaclav Kotesovec, Mar 31 2018

A299209 Expansion of 1/(1 - xProduct_{k>=1} (1 - kx^k)).

Original entry on oeis.org

1, 1, 0, -3, -6, -5, 11, 37, 59, 13, -155, -402, -415, 263, 1981, 3748, 2289, -6643, -22642, -31322, -187, 99040, 229410, 216823, -230029, -1223267, -2097812, -955237, 4468902, 13393758, 16752461, -3891704, -62382597, -131974181, -106680562, 173622424, 741553622, 1163057561, 329176545

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

N. J. A. Sloane, Transforms

Antidiagonal sums of A297323.

Cf. A022661, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299210, A299211, A299212.

nmax = 38; CoefficientList[Series[1/(1 - x Product[1 - k x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 - k*x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A022661(k-1)*a(n-k).

A299210 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 + kx^k)).

Original entry on oeis.org

1, 1, 0, -2, -5, -3, 5, 20, 27, 17, -53, -152, -192, 31, 576, 1110, 694, -1297, -4519, -6160, -1107, 13665, 31914, 30643, -19339, -119260, -196142, -103318, 289543, 859631, 1062684, 13710, -2690348, -5675946, -4940757, 4167527, 21343918, 33874107, 16524162, -51704908, -150454546

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

N. J. A. Sloane, Transforms

Antidiagonal sums of A297325.

Cf. A022693, A067687, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299211, A299212.

nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + k x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A022693(k-1)*a(n-k).

A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

Original entry on oeis.org

1, 1, 0, -2, -5, -4, 4, 21, 35, 23, -47, -165, -239, -78, 479, 1273, 1508, -138, -4429, -9451, -8845, 6207, 37937, 67123, 45144, -83355, -308078, -455109, -166872, 873799, 2393041, 2916869, -73472, -8133572, -17828640, -17294146, 10383571, 70275162, 127401305, 90368779, -147825714

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

N. J. A. Sloane, Transforms

Antidiagonal sums of A279928.

Cf. A067687, A255528, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299211.

nmax = 40; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

a(0) = 1; a(n) = Sum_{k=1..n} A255528(k-1)*a(n-k).

A318581 Expansion of 1/(1 + x*Product_{k>=1} 1/(1 - x^k)).

Original entry on oeis.org

1, -1, 0, -1, 0, -1, 1, -1, 3, -1, 5, -2, 7, -7, 9, -16, 11, -29, 20, -46, 45, -66, 94, -95, 175, -161, 294, -307, 458, -594, 715, -1096, 1193, -1891, 2132, -3106, 3916, -5063, 7083, -8484, 12347, -14770, 20867, -26310, 34898, -46771, 58967, -81665, 101680, -139951, 178094, -237620

Offset: 0

Ilya Gutkovskiy, Aug 29 2018

sign

			G.f. = 1 - x - x^3 - x^5 + x^6 - x^7 + 3*x^8 - x^9 + 5*x^10 - 2*x^11 + 7*x^12 - 7*x^13 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318582, A331484.

Cf. A000041, A010815.

seq(coeff(series((1+x*mul((1-x^k)^(-1),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 55); # Muniru A Asiru, Aug 30 2018

nmax = 51; CoefficientList[Series[1/(1 + x Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[PartitionsP[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 51}]

G.f.: 1/(1 + x*Sum_{k>=0} A000041(k)*x^k).

a(0) = 1; a(n) = -Sum_{k=1..n} A000041(k-1)*a(n-k).

A318582 Expansion of 1/(1 + x*Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, 0, 1, 0, 0, 0, 1, -1, 0, 1, -3, 2, -1, -3, 4, -4, 0, 3, -5, 4, 0, -2, 4, -1, 1, 0, 3, -2, 0, 6, -11, 9, -1, -13, 18, -17, 1, 13, -23, 17, -4, -8, 13, -8, 7, -6, 15, -10, -3, 33, -50, 42, 0, -56, 85, -72, 6, 59, -100, 75, -23, -34, 53, -44, 35

Offset: 0

Ilya Gutkovskiy, Aug 29 2018

sign

			G.f. = 1 - x - x^4 + x^5 - x^6 + x^8 - x^9 + x^10 + x^13 + x^17 - x^18 + x^20 - 3*x^21 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A331484.

Cf. A000009, A081362.

a:=series(1/(1+x*mul(1+x^k,k=1..100)),x=0,76): seq(coeff(a,x,n),n=0..75); # Paolo P. Lava, Apr 02 2019

nmax = 75; CoefficientList[Series[1/(1 + x Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[PartitionsQ[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

G.f.: 1/(1 + x*Sum_{k>=0} A000009(k)*x^k).

a(0) = 1; a(n) = -Sum_{k=1..n} A000009(k-1)*a(n-k).

A331484 Expansion of 1/(1 + x*Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, -1, 2, -2, 3, -3, 3, -2, -1, 5, -13, 22, -36, 51, -68, 82, -86, 75, -31, -52, 201, -421, 732, -1125, 1575, -2024, 2344, -2370, 1807, -327, -2532, 7210, -14128, 23486, -35027, 47799, -59594, 66717, -63246, 41012, 10696, -104335, 252653, -465825, 746343

Offset: 0

Seiichi Manyama, Jan 18 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582.

Cf. A010815.

m = 44; CoefficientList[Series[1/(1 + x*Product[1 - x^k, {k, 1, m}]), {x, 0, m}], x] (* Amiram Eldar, May 05 2021 *)

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

a(0) = 1, a(n) = -Sum_{k=1..n} A010815(k-1)*a(n-k) for n > 0.

A317536 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 4, 6, 10, 18, 30, 50, 86, 145, 245, 417, 705, 1193, 2024, 3427, 5804, 9836, 16660, 28220, 47811, 80991, 137197, 232423, 393729, 666982, 1129898, 1914078, 3242495, 5492898, 9305130, 15763154, 26703273, 45236138, 76631348, 129815818, 219911870, 372537244, 631089250

Offset: 0

Ilya Gutkovskiy, Jul 30 2018

nonn

Invert transform of A111133.

Alois P. Heinz, Table of n, a(n) for n = 0..4371
N. J. A. Sloane, Transforms

Cf. A000009, A050342, A089259, A111133, A270995, A279785, A299106, A304969, A305651, A317535.

seq(coeff(series(1/(1+1/(1-x)-mul(1+x^k,k=1..n)), x,n+1),x,n),n=0..50); # Muniru A Asiru, Jul 30 2018
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*(b(i)-1), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Dec 16 2022

nmax = 42; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[(PartitionsQ[k] - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(PartitionsQ[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

G.f.: 1/(1 - Sum_{k>=1} A111133(k)*x^k).

A307067 Expansion of 1/(2 - Product_{k>=2} (1 + x^k)).

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 6, 12, 19, 36, 60, 108, 187, 328, 576, 1005, 1765, 3084, 5408, 9461, 16575, 29017, 50812, 88977, 155792, 272813, 477684, 836466, 1464654, 2564685, 4490833, 7863610, 13769463, 24110774, 42218847, 73926591, 129448088, 226667986, 396903536, 694991728

Offset: 0

Ilya Gutkovskiy, Mar 22 2019

nonn

Invert transform of A025147.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A025147, A055887, A299106, A304969, A307057, A307060, A317536.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1+x^j: j in [2..m+2]])) )); // G. C. Greubel, Jan 24 2024

a:=series(1/(2-mul((1+x^k),k=2..100)),x=0,40): seq(coeff(a,x,n),n=0..39); # Paolo P. Lava, Apr 03 2019

nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^k), {k, 2, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/( 2 - product(1+x^j for j in range(2, m+3)) )
def A307067_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307067_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A025147(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: (1+x)/(2*(1+x) - QP(x^2)/QP(x)), where QP(x) = QPochhammer(x).
G.f.: (1+x)/(2*(1+x) - x^(1/24)*eta(x^2)/eta(x)), where eta(x) is the Dedekind eta function. (End)

A302020 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k))/(1 - x^(2*k-1))).

Original entry on oeis.org

1, 1, 2, 5, 12, 28, 66, 156, 367, 863, 2031, 4779, 11244, 26456, 62248, 146462, 344608, 810822, 1907769, 4488757, 10561519, 24850017, 58469179, 137571128, 323688747, 761601701, 1791959579, 4216270956, 9920391613, 23341519267, 54919860316, 129219997322, 304039515247, 715369360371

Offset: 0

Ilya Gutkovskiy, Mar 30 2018

nonn

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Partition Function b_k

Antidiagonal sums of A296068.

Cf. A001935, A299106, A299108.

nmax = 33; CoefficientList[Series[1/(1 - x Product[(1 + x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 + (1 - x) QPochhammer[-1, x^2]/(2 QPochhammer[1/x, x^2])), {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[1/(1 - x EllipticTheta[2, 0, x]/(Sqrt[2] x^(1/8) EllipticTheta[2, Pi/4, Sqrt[x]])), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(4*k))/(1 - x^k)).
G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k))).
a(0) = 1; a(n) = Sum_{k=1..n} A001935(k-1)*a(n-k).

cp's OEIS Frontend

A302017 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^(2*k-1))).

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A299209 Expansion of 1/(1 - x*Product_{k>=1} (1 - k*x^k)).

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A299210 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + k*x^k)).

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A299212 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)^k).

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A318581 Expansion of 1/(1 + x*Product_{k>=1} 1/(1 - x^k)).

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A318582 Expansion of 1/(1 + x*Product_{k>=1} (1 + x^k)).

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A331484 Expansion of 1/(1 + x*Product_{k>=1} (1 - x^k)).

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A317536 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} (1 + x^k)).

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A302017 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k-1))).

A299209 Expansion of 1/(1 - xProduct_{k>=1} (1 - kx^k)).

A299210 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 + kx^k)).

A302020 Expansion of 1/(1 - xProduct_{k>=1} (1 + x^(2k))/(1 - x^(2*k-1))).