A299211 -id:A299211 - cp's OEIS frontend

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

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).

A276554 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)^(j*k) in powers of x.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -2, 0, 1, -3, -3, -1, 0, 1, -4, -3, 2, 0, 0, 1, -5, -2, 8, 6, 4, 0, 1, -6, 0, 16, 12, 12, 4, 0, 1, -7, 3, 25, 13, 9, 1, 7, 0, 1, -8, 7, 34, 5, -12, -29, -10, 3, 0, 1, -9, 12, 42, -15, -51, -78, -54, -32, -2, 0, 1, -10, 18, 48, -49, -102

Offset: 0

Seiichi Manyama, Apr 10 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -2, -3, -3, -2, ...
   0, -1,  2,  8, 16, ...
   0,  0,  6, 12, 13, ...

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

Columns k=0-5 give: A000007, A073592, A276551, A276552, A316463, A316464.
Main diagonal gives A281267.
Antidiagonal sums give A299211.
Cf. A255961, A277938.

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

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).

cp's OEIS Frontend

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

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A276554 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)^(j*k) in powers of x.

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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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Author

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Programs

Mathematica

Formula

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

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Author

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Programs

Mathematica

Formula

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

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