A299108 -id:A299108 - 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

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

Original entry on oeis.org

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).

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).

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).

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).

cp's OEIS Frontend

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

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A299105 Expansion of 1/(1 - x*Product_{k>=1} (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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Maple

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

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

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Maple

Mathematica

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

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

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