A010822 -id:A010822 - cp's OEIS frontend

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.

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, ...].

A322043 Numbers k such that the coefficient of x^k in the expansion of Product_{m >= 1} (1-x^m)^15 is zero.

Original entry on oeis.org

53, 482, 1340, 2627, 4343, 6488, 9062, 12065, 15497, 19358, 23648, 28367, 33515, 39092, 45098, 51533, 58397, 65690, 73412, 81563, 90143, 99152, 108590, 118457, 128753, 139478, 150632, 162215, 174227, 186668, 199538, 212837, 226565, 240722, 255308, 270323, 285767, 301640, 317942, 334673, 351833, 369422, 387440

Offset: 1

N. J. A. Sloane, Dec 07 2018

nonn

Van der Blij, discussing the conjecture that the Ramanujan numbers tau(k) (see A000594) are never zero, mentions that a certain "Ferguson" had shown that 52 is a member of the current sequence. No details were given, and the 52 appears to be a typo for 53.

The coefficients of the expansion of Product_{m >= 1} (1-x^m)^15 are given in A010822.

F. Van der Blij, "The function tau(n) of S. Ramanujan (an expository lecture)." Math. Student 18 (1950): 83-99. See page 85.

Joerg Arndt, Table of n, a(n) for n = 1..216

Cf. A000594, A010822, A302057.

sigma[k_] := sigma[k] = DivisorSigma[1, k];
a[0] = 1; a[n_] := a[n] = -15/n Sum[sigma[k] a[n-k], {k, 1, n}];
Reap[For[k = 1, k <= 200000, k++, If[a[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 20 2018 *)

/* start with sufficient memory, e.g., gp -s16G */
x='x+O('x^1000000); v=Vec(eta(x)^15 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ Joerg Arndt, Dec 20 2018

Conjectures from Colin Barker, Dec 07 2018: (Start)
G.f.: x*(53 + 323*x + 53*x^2) / (1 - x)^3.
a(n) = (429*n^2 - 429*n + 106) / 2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
(End)
E.g.f.: (1/2)*exp(x)*(106 + 858*x + 429*x^2). - conjectured by Stefano Spezia, Dec 07 2018 after the conjectures of Colin Barker

a(4)-a(7) supplied by Rémy Sigrist, Dec 07 2018, from the b-file for A010822.
a(8)-a(19) from Seiichi Manyama, Dec 07 2018
a(20)-a(31) from Jean-François Alcover, Dec 20 2018
More terms from Joerg Arndt, Dec 20 2018

cp's OEIS Frontend

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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