A299208 -id:A299208 - cp's OEIS frontend

A299105 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)).

1, 1, 0, -2, -3, -1, 5, 10, 7, -9, -29, -30, 10, 77, 108, 22, -184, -351, -207, 372, 1041, 969, -516, -2835, -3655, -284, 6990, 12190, 5977, -14957, -37044, -30994, 24144, 103374, 122409, -7715, -262704, -420585, -162274, 589068, 1309674, 972747, -1057935, -3742955

Offset: 0

Ilya Gutkovskiy, Feb 02 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286354.
Cf. similar sequences: A067687, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A010815, A299108.

nmax = 43; CoefficientList[Series[1/(1 - x Product[1 - x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[1/(1 - x QPochhammer[x, x]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A010815(k-1)*a(n-k).

A286352 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 1, -1, 0, 1, -4, 3, -2, 1, 0, 1, -5, 6, -4, 4, -1, 0, 1, -6, 10, -8, 9, -4, 1, 0, 1, -7, 15, -15, 17, -12, 5, -1, 0, 1, -8, 21, -26, 30, -28, 15, -6, 2, 0, 1, -9, 28, -42, 51, -56, 38, -21, 9, -2, 0, 1, -10, 36, -64, 84

Offset: 0

Seiichi Manyama, May 08 2017

			Square array begins:
   1,  1,  1,  1,  1,   1, ...
   0, -1, -2, -3, -4,  -5, ...
   0,  0,  1,  3,  6,  10, ...
   0, -1, -2, -4, -8, -15, ...
   0,  1,  4,  9, 17,  30, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-32 give: A000007, A081362, A022597-A022627.
Main diagonal gives A255526.
Antidiagonal sums give A299208.
Cf. A286335.

G.f. of column k: Product_{j>=1} 1/(1 + x^j)^k.

A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

Original entry on oeis.org

1, 1, 0, -3, -6, -4, 12, 39, 52, -9, -186, -392, -285, 610, 2291, 3200, -150, -10626, -23487, -18841, 32957, 134848, 198246, 13961, -605248, -1409604, -1234474, 1744213, 7898753, 12209679, 2161666, -34344627, -84393284, -79993042, 90692470, 461463974, 749309529, 207447895, -1939084232

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

Robert Israel, Table of n, a(n) for n = 0..3925
N. J. A. Sloane, Transforms

Antidiagonal sums of A276554.

Cf. A067687, A073592, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299212.

N:= 100: # for a(0)..a(N)
S:= series(1/(1-x*mul((1-x^k)^k,k=1..N)),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Feb 05 2023

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

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

a(0) = 1; a(n) = Sum_{k=1..n} A073592(k-1)*a(n-k).

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

Original entry on oeis.org

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

A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, -1, 1, -2, 4, -7, 12, -21, 38, -68, 120, -212, 377, -670, 1188, -2107, 3740, -6638, 11778, -20898, 37084, -65808, 116775, -207212, 367696, -652478, 1157815, -2054524, 3645730, -6469316, 11479734, -20370656, 36147506, -64143372, 113821732, -201975429, 358403220, -635982680, 1128544452, -2002589998

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

sign

Invert transform of A081362.

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

Cf. A081362, A299208, A304969, A307058.

Cf. A307057, A307058, A307059, A307062, A307063.

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

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

m=80;
def f(x): return 1/( 2 - product(1-x^(2*j-1) for j in range(1,m+3)) )
def A307060_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307060_list(m) # G. C. Greubel, Jan 24 2024

G.f.: 1/(2 - Product_{k>=1} (1 - x^(2*k-1))).
a(0) = 1; a(n) = Sum_{k=1..n} A081362(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: 1/(2 - QPochhammer(x)/QPochhammer(x^2)).
G.f.: 1/(2 - x^(1/24)*eta(x)/eta(x^2)), where eta(x) is the Dedekind eta function. (End)

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).

cp's OEIS Frontend

A299105 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)).

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A286352 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + x^j)^k.

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A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

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Maple

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

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A307060 Expansion of 1/(2 - Product_{k>=1} 1/(1 + x^k)).

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Magma

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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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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)).