A276551 -id:A276551 - cp's OEIS frontend

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.

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).

A276552 Expansion of Product_{k>0} (1 - x^k)^(k*3).

Original entry on oeis.org

1, -3, -3, 8, 12, 9, -29, -54, -51, 8, 168, 273, 270, -18, -546, -1220, -1539, -969, 796, 3693, 6591, 8098, 5412, -2568, -16053, -31524, -42195, -38684, -11868, 41994, 117630, 193365, 235497, 197758, 42852, -247224, -639547, -1041432, -1291425, -1184100, -520650

Offset: 0

Seiichi Manyama, Apr 10 2017

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

Column k=3 of A276554.

Product_{k>0} (1-x^k)^(k*m): A255610 (m=-3), A073592 (m=1), A276551 (m=2), this sequence (m=3).

A319669 Expansion of Product_{k>=1} (1 - x^k)^(2*k-1).

Original entry on oeis.org

1, -1, -3, -2, 1, 10, 13, 15, -1, -30, -63, -89, -80, -14, 131, 304, 493, 561, 434, -32, -836, -1895, -2960, -3583, -3240, -1338, 2401, 8004, 14499, 20494, 23369, 20401, 8567, -13741, -46408, -85717, -124027, -149612, -147167, -101002, 2520, 168026, 388077, 634914

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

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Cf. A000203, A001157, A073592, A253289, A255835, A276551, A278945, A285069.

a:=series(mul((1-x^k)^(2*k-1),k=1..100),x=0,44): seq(coeff(a,x,n),n=0..43); # Paolo P. Lava, Apr 02 2019

nmax = 43; CoefficientList[Series[Product[(1 - x^k)^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 43; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 2 DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (1 - 2 d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

G.f.: exp(Sum_{k>=1} (sigma_1(k) - 2*sigma_2(k))*x^k/k).

cp's OEIS Frontend

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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A276552 Expansion of Product_{k>0} (1 - x^k)^(k*3).

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A319669 Expansion of Product_{k>=1} (1 - x^k)^(2*k-1).

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