A299105 -id:A299105 - cp's OEIS frontend

A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)).

1, 1, 2, 4, 9, 19, 41, 88, 189, 405, 869, 1864, 3998, 8575, 18392, 39448, 84610, 181475, 389235, 834848, 1790617, 3840591, 8237462, 17668057, 37895195, 81279216, 174331098, 373912708, 801983781, 1720128713, 3689404772, 7913191304, 16972547194, 36403436640

Offset: 0

Ilya Gutkovskiy, Feb 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286335.

Cf. A000009, A067687, A299105, A299108.

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

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A000009(k-1)*a(n-k).
a(n) ~ c * d^n, where d = 2.14484226934608840026733598736202689102117985119507858808036465196716739... is the root of the equation QPochhammer(1/d, 1/d^2)*d = 1 and c = 0.4217892515709863296976217395517853732959704351198250451894928058439... = 2/(2+Derivative[0, 1][QPochhammer][-1, 1/d]/d^2). - Vaclav Kotesovec, Feb 03 2018, updated Mar 31 2018

A286354 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, 0, 0, 1, -4, 0, 2, 0, 0, 1, -5, 2, 5, 1, 1, 0, 1, -6, 5, 8, 0, 2, 0, 0, 1, -7, 9, 10, -5, 0, -2, 1, 0, 1, -8, 14, 10, -15, -4, -7, 0, 0, 0, 1, -9, 20, 7, -30, -6, -10, 0, -2, 0, 0, 1, -10, 27, 0, -49, 0, -5, 8, 0, -2, 0, 0, 1, -11, 35, -12, -70, 21, 11, 25, 9, 0, 1, 0, 0

Offset: 0

Ilya Gutkovskiy, May 08 2017

A(n,k) number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts with k types of each part.

			A(3,2) = 2 because we have [2, 1], [2', 1], [2, 1'], [2', 1'] (number of partitions of 3 into an even number of distinct parts with 2 types of each part), [3], [3'] (number of partitions of 3 into an odd number of distinct parts with 2 types of each part) and 4 - 2 = 2.
Square array begins:
1,  1,  1,  1,  1,   1,  ...
0, -1, -2, -3, -4,  -5,  ...
0, -1, -1,  0,  2,   5,  ...
0,  0,  2,  5,  8,  10,  ...
0,  0,  1,  0, -5, -15,  ...
0,  1,  2,  0, -4,  -6,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-20 give: A000007, A010815, A002107, A010816, A000727, A000728, A000729, A000730, A000731, A010817, A010818, A010819, A000735, A010820, A010821, A010822, A000739, A010823, A010824, A010825, A010826.
Main diagonal gives A008705.
Antidiagonal sums give A299105.

A:= proc(n, k) option remember; `if`(n=0, 1, -k*
      add(numtheory[sigma](j)*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jun 21 2018

Table[Function[k, SeriesCoefficient[Product[(1 - x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x, x, Infinity]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[(-1)^i*x^(i*(3*i + 1)/2), {i, -Infinity, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 - x^j)^k.
G.f. of column k: (Sum_{j=-inf..inf} (-1)^j*x^(j*(3*j+1)/2))^k.
Column k is the Euler transform of period 1 sequence [-k, -k, -k, ...].

A299108 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).

Original entry on oeis.org

1, 1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16805, 49071, 143289, 418411, 1221781, 3567663, 10417761, 30420401, 88829145, 259385701, 757419669, 2211704625, 6458291945, 18858546645, 55067931981, 160801210705, 469547855419, 1371104033121, 4003694720243

Offset: 0

Ilya Gutkovskiy, Feb 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Antidiagonal sums of A288515.

Cf. A015128, A032803, A067687, A299105, A299106.

S:= series(1/(1-x/JacobiTheta4(0,x)),x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Feb 02 2018

nmax = 28; CoefficientList[Series[1/(1 - x Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x/EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x QPochhammer[-x, x]/QPochhammer[x, x]), {x, 0, nmax}], x]

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).
G.f.: 1/(1 - x/theta_4(x)), where theta_4() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A015128(k-1)*a(n-k).
a(n) ~ c * d^n, where d = 2.9200517419026569743994130834319365190407162724411912701937027582419975778... is the root of the equation EllipticTheta(4, 0, 1/d) * d = 1 and c = 0.372842695601022868809531452599286285949969156503576039087883242107... = 2*Log[r]*QPochhammer[r] / (2*QPochhammer[r] * (Log[1 - r] + Log[r] + QPolyGamma[1, r]) + r*Log[r] * (r * Derivative[0, 1][QPochhammer][-1, r] - 2*Derivative[0, 1][QPochhammer][r, r])), where r = 1/d. Equivalently, c = EllipticTheta[4, 0, r]^2 / (r *(EllipticTheta[4, 0, r] - r * Derivative[0, 0, 1][EllipticTheta][4, 0, r])). - Vaclav Kotesovec, Feb 03 2018, updated Mar 31 2018

A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

Original entry on oeis.org

1, 1, 0, -1, -2, -1, 1, 3, 3, 1, -3, -6, -5, 1, 9, 12, 5, -9, -20, -18, 1, 26, 38, 21, -21, -61, -62, -9, 72, 120, 81, -44, -177, -205, -64, 186, 366, 293, -63, -496, -657, -304, 445, 1084, 1014, 33, -1341, -2053, -1238, 959, 3132, 3378, 770, -3474, -6260, -4619, 1656, 8809, 10929, 4306, -8520

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Antidiagonal sums of A286352.
Cf. similar sequences: A067687, A299105, A299106, A299208, A302017, A318581, A318582, A331484.
Cf. A081362, A299108, A299162, A299164, A299166, A299167, A299209, A299210, A299211, A299212.

nmax = 60; CoefficientList[Series[1/(1 - x Product[1/(1 + x^k), {k, 1, nmax}]), {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} A081362(k-1)*a(n-k).

A299162 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 - kx^k)).

Original entry on oeis.org

1, 1, 2, 6, 17, 49, 135, 380, 1051, 2925, 8119, 22548, 62574, 173767, 482360, 1339126, 3717700, 10321163, 28653557, 79548612, 220843925, 613110573, 1702128034, 4725475979, 13118945083, 36421037100, 101112695940, 280710759278, 779313926949, 2163544401343, 6006468273440

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2256
N. J. A. Sloane, Transforms

Antidiagonal sums of A297328.

Cf. A006906, A067687, A299105, A299106, A299108, A299164, A299166, A299167.

nmax = 30; 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} A006906(k-1)*a(n-k).

A299164 Expansion of 1/(1 - xProduct_{k>=1} (1 + kx^k)).

Original entry on oeis.org

1, 1, 2, 5, 14, 35, 91, 233, 597, 1517, 3885, 9922, 25333, 64683, 165181, 421828, 1077277, 2750993, 7025168, 17940298, 45814165, 116996152, 298774246, 762982615, 1948434235, 4975732669, 12706571546, 32448880807, 82864981016, 211613009498, 540397935771, 1380018797044, 3524165721799

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Antidiagonal sums of A297321.

Cf. A022629, A067687, A299105, A299106, A299108, A299162, A299166, A299167.

nmax = 32; 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} A022629(k-1)*a(n-k).

A299166 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).

Original entry on oeis.org

1, 1, 2, 6, 17, 48, 132, 365, 1003, 2759, 7583, 20843, 57283, 157442, 432719, 1189317, 3268818, 8984318, 24693343, 67869557, 186539251, 512702559, 1409161449, 3873076007, 10645137706, 29258128633, 80415877302, 221022792843, 607480469466, 1669658209311, 4589050472041

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Antidiagonal sums of A255961.

Cf. A000219, A067687, A299105, A299106, A299108, A299162, A299164, A299167.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
       b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(n-j, j), j=0..n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 04 2018

nmax = 30; 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} A000219(k-1)*a(n-k).

A299167 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)^k).

Original entry on oeis.org

1, 1, 2, 5, 14, 36, 94, 243, 628, 1619, 4178, 10776, 27793, 71682, 184879, 476832, 1229830, 3171942, 8180989, 21100215, 54421187, 140361900, 362018270, 933709453, 2408202606, 6211182512, 16019743522, 41317765457, 106565859669, 274852289679, 708892898170, 1828360759013, 4715667307920

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

N. J. A. Sloane, Transforms

Antidiagonal sums of A277938.

Cf. A026007, A067687, A299105, A299106, A299108, A299162, A299164, A299166.

nmax = 32; 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} A026007(k-1)*a(n-k).

A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, -1, 0, 1, -1, 1, -1, 1, 0, -2, 4, -4, 1, 3, -5, 4, -3, 3, -1, -6, 13, -12, 2, 9, -13, 10, -6, 6, -4, -9, 28, -30, 5, 25, -28, 5, 9, 7, -27, 11, 32, -47, 2, 51, -27, -74, 128, -34, -131, 183, -78, -15, -37, 97, 89, -480, 649, -242, -498, 904, -663, 223, -140, 169, 488, -1818

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

sign

Invert transform of A010815.

Alternating row sums of Riordan triangle (1, 1 - Product_{j>=1} (1-x^j) ), See A341418(n, m) without column {1, repeat(0)} for m = 0 and n >= 0. - Wolfdieter Lang, Feb 17 2021

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A010815, A055887, A299105, A341418.

Cf. A307057, A307058, A307060, A307062, A307063.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 - x^j: j in [1..m+2]])) )); // G. C. Greubel, Sep 08 2023

nmax=65; CoefficientList[Series[1/(2 - Product[(1 - x^k), {k, nmax}]), {x, 0, nmax}], x]

from sage.modular.etaproducts import qexp_eta
m=80;
def f(x): return 1/(2 - qexp_eta(QQ[['q']], m+2).subs(q=x) )
def A307059_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307059_list(m) # G. C. Greubel, Sep 08 2023

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

G.f.: 1/(2 - QPochhammer(x)). - G. C. Greubel, Sep 08 2023

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).

cp's OEIS Frontend

A299106 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)).

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A286354 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)^k.

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A299108 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).

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A299208 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 + x^k)).

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A299162 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - k*x^k)).

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A299164 Expansion of 1/(1 - x*Product_{k>=1} (1 + k*x^k)).

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A299166 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).

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A299167 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)^k).

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A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).

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A299162 Expansion of 1/(1 - xProduct_{k>=1} 1/(1 - kx^k)).

A299164 Expansion of 1/(1 - xProduct_{k>=1} (1 + kx^k)).