A010818 -id:A010818 - 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, ...].

A322431 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^10 is zero.

Original entry on oeis.org

6, 13, 17, 27, 28, 34, 36, 39, 41, 48, 55, 59, 61, 62, 72, 74, 76, 82, 83, 90, 93, 94, 97, 104, 105, 111, 112, 116, 121, 125, 127, 128, 131, 132, 138, 139, 146, 149, 151, 152, 153, 160, 168, 169, 174, 181, 182, 183, 188, 193, 195, 197, 202, 204, 207, 209, 211, 214, 215

Offset: 1

Seiichi Manyama, Dec 07 2018

nonn

Indices of zero entries in A010818.

Also: numbers k such that 24k + 10 cannot be written as (12m+3)^2 + (4n+1)^2 with integers m, n. In this case, 12k + 5 is never prime. - M. F. Hasler, Jun 30 2025

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322430 (m=8), this sequence (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

my(x='x+O('x^300)); Vec(select(x->(x==0), Vec(eta(x)^10 - 1), 1)) \\ Michel Marcus, Dec 08 2018

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

A122266 Expansion of f(-q)^2 * Q(q) in powers of q.

Original entry on oeis.org

1, 238, 1679, 2162, 2401, -6958, -1442, -23040, 1918, -9362, 14641, 0, 80640, -20398, 28083, 64078, -38398, -69120, 0, -90482, -58562, 0, -241920, 100558, 146879, 0, -193438, 399602, 104638, 114002, 130321, 24242, 0, 107282, -276962, 351118

Offset: 0

Michael Somos, Aug 28 2006

sign

f(-q) (g.f. A010815) and Q(q) (g.f. A004009) are Ramanujan q-series.

			G.f. = 1 + 238*x + 1679*x^2 + 2162*x^3 + 2401*x^4 - 6958*x^5 - 1442*x^6 + ...
G.f. = q + 238*q^13 + 1679*q^25 + 2162*q^37 + 2401*q^49 - 6958*q^61 - 1442*q^73 + ...

Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

Cf. A002107, A004009, A010818.

a[ n_] := SeriesCoefficient[ (1 + 240 Sum[ DivisorSigma[ 3, k] q^k, {k, n}]) QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jun 24 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 (EllipticTheta[ 4, 0, q]^8 + EllipticTheta[ 2, 0, q^(1/2)]^8), {q, 0, n}]; (* Michael Somos, Jun 25 2013 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * sum( k=1, n, 240 * sigma( k, 3) * x^k, 1 + A), n))};

{a(n) = my(A, B); if( n<0, 0, A = x * O(x^n); B = 64 * x * (eta(x^4 + A) / eta(x + A))^8; polcoeff( (1 + 4*B + B^2) * eta(x + A)^18 / eta(x^2 + A)^8, n))}; /* Michael Somos, Jun 25 2013 */

{a(n) = my(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 12*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p<5, 0, if( p%12 > 6, if( e%2, 0, p^(2*e)), forstep( i = 1, sqrtint( p), 2, if( issquare( p - i^2, &y), x=i; break)); if( p%12 == 5, a1 = 8 * x*y * (x-y) * (x+y) * (-1)^((x%6==1) + (y%6==4)), a1 = 2 * (x^2-y^2+2*x*y) * (x^2-y^2-2*x*y) * (-1)^(x%6==3) ); a0 = 1; y = a1; for( i=2, e, x = y * a1 - p^4 * a0; a0=a1; a1=x); a1 )))))}; /* Michael Somos, Jun 24 2013 */

Expansion of q^(-1/12) * (eta(q)^16 + 256 * eta(q)^8 * eta(q^4)^8 + 4096 * eta(q^4)^16) * eta(q)^2 / eta(q^2)^8 in powers of q. - Michael Somos, Jun 25 2013
a(n) = b(12*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 7 or 11 (mod 12), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 or 5 (mod 12). - Michael Somos, Jun 24 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12^5 (t/i)^5 f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 06 2014
Convolution of A002107 and A004009. - Michael Somos, Jun 25 2013
Expansion of q^(-1/12) * (eta(q)^24 + 256*eta(q^2)^24) / (eta(q)^6*eta(q^2)^8) = q^(-1/12) * (eta(q)^12 + 250*eta(q)^6*eta(q^5)^6 + 3125*eta(q^5)^12) / eta(q^5)^2 in powers of q. - Michael Somos, Feb 03 2023

A234565 Expansion of f(-q^3)^2 * Q(q^3) + 48 * q * f(-q^3)^10 in powers of q.

Original entry on oeis.org

1, 48, 0, 238, -480, 0, 1679, 1680, 0, 2162, -1440, 0, 2401, -5040, 0, -6958, 11424, 0, -1442, 0, 0, -23040, -12480, 0, 1918, -7920, 0, -9362, 6720, 0, 14641, 50592, 0, 0, -36960, 0, 80640, -28560, 0, -20398, 0, 0, 28083, -34320, 0, 64078, 103776, 0, -38398

Offset: 0

Michael Somos, Jan 06 2014

sign

f(-q) (g.f. A010815) and Q(q) (g.f. A004009) are Ramanujan q-series.

			G.f. = 1 + 48*x + 238*x^3 - 480*x^4 + 1679*x^6 + 1680*x^7 + 2162*x^9 + ...
G.f. = q + 48*q^5 + 238*q^13 - 480*q^17 + 1679*q^25 + 1680*q^29 + 2162*q^37 + ...

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

Cf. A010818, A122266.

eta[q_]:= q^(1/24)*QPochhammer[q]; Q:= (eta[q^3]^24 + 256*eta[q^6]^24)/( eta[q^3]*eta[q^6])^8; a:= CoefficientList[Series[q^(-1/4)*eta[q^3]^2*(48*q^(0/4)*eta[q^3]^8 + Q), {q, 0, 55}], q]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = local(A, B); if( n<0, 0, A = x * O(x^n); B = 64 * x^3 * (eta(x^12 + A) / eta(x^3 + A))^8; polcoeff( 48 * x * eta(x^3 + A)^10 + (1 + 4*B + B^2) * eta(x^3 + A)^18 / eta(x^6 + A)^8, n))}

{a(n) = local(A, p, e, i, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p<5, 0, if( p%12 > 6, if( e%2, 0, p^(2*e)), forstep( i = 1, sqrtint( p), 2, if( issquare( p - i^2, &y), x=i; break)); if( p%12 == 5, a1 = 8 * x*y * (x-y) * (x+y) * (-1)^((x%6==1) + (y%6==4)), a1 = 2 * (x^2-y^2+2*x*y) * (x^2-y^2-2*x*y) * (-1)^(x%6==3) ); a0 = 1; y = a1; for( i=2, e, x = y * a1 - p^4 * a0; a0=a1; a1=x); a1 )))))}

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12^5 (t/i)^5 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 7 or 11 (mod 12), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 or 5 (mod 12).
a(3*n + 2) = 0. a(3*n) = A122266(n). a(3*n + 1) = 48 * A010818(n).

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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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A322431 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1-x^j)^10 is zero.

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A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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A122266 Expansion of f(-q)^2 * Q(q) in powers of q.

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A234565 Expansion of f(-q^3)^2 * Q(q^3) + 48 * q * f(-q^3)^10 in powers of q.

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