A286354 -id:A286354 - cp's OEIS frontend

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

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).

A008705 Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n.

Original entry on oeis.org

1, -1, -1, 5, -5, -6, 11, 41, -125, -85, 1054, -2069, -209, 8605, -15625, 3990, 14035, 36685, -130525, -254525, 1899830, -3603805, -134905, 13479425, -25499225, 23579969, -64447293, 237487433, -133867445, -1795846200, 6309965146, -6788705842, -11762712973

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

sign

Degree of resulting polynomial is A002411(n). - Michel Marcus, Sep 05 2013
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k. - Peter Bala, Jan 31 2022
Conjectures: the supercongruences a(p) == -1 - p (mod p^2) and a(2*p) == p - 1 (mod p^2) hold for all primes p >= 3. - Peter Bala, Apr 18 2023

			(1-x)^1 = -x + 1, hence a(1) = -1.
(1-x^2)^2*(1-x)^2 = x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1, hence a(2) = -1.

Seiichi Manyama, Table of n, a(n) for n = 0..2856 (terms 0..256 from N. J. A. Sloane)
Morris Newman, Further identities and congruences for the coefficients of modular forms, Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.
Morris Newman, Further identities and congruences for the coefficients of modular forms [annotated scanned copy], Canadian J. Math 10 (1958): 577-586. See Table 1, column p=5.

Bisections: A262308, A262309.

Main diagonal of A286354.

C5:=proc(r) local t1,n; t1:=mul((1-x^n)^r,n=1..r+2); series(t1,x,r+1); coeff(%,x,r); end;
[seq(C5(i),i=0..30)]; # N. J. A. Sloane, Oct 04 2015
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, -k*
      add(numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 21 2018

With[{m = 40}, Table[SeriesCoefficient[Series[(Product[1-x^j, {j, n}])^n, {x, 0, m}], n], {n, 0, m}]] (* G. C. Greubel, Sep 09 2019 *)

a(n) = polcoeff(prod(m = 1, n, (1-x^m)^n), n); \\ Michel Marcus, Sep 05 2013

a(n) = [x^n] exp(-n*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 30 2018

More terms from Michel Marcus, Sep 05 2013

a(0)=1 prepended by N. J. A. Sloane, Oct 04 2015

A360401 a(n) = A356133(A360393(n)).

Original entry on oeis.org

2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, 124, 133, 146, 151, 164, 173, 178, 191, 197, 206, 218, 227, 233, 242, 253, 260, 272, 280, 287, 295, 308, 317, 322, 335, 341, 350, 362, 371, 377, 385, 398, 403, 415, 425, 430, 443, 449, 457, 470

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286355, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A010840 Expansion of Product_{k>=1} (1-x^k)^40.

Original entry on oeis.org

1, -40, 740, -8320, 62530, -322048, 1085240, -1799680, -2821065, 26012480, -66837420, 35093760, 268749870, -783902720, 526221400, 1691816960, -3960854625, 1042577120, 5103133240, -380798080, -10159511430

Offset: 0

N. J. A. Sloane

sign

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Column k=40 of A286354.

Cf. A000203.

a(0) = 1, a(n) = -(40/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

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))

A360400 a(n) = A356133(A360392(n)).

Original entry on oeis.org

7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, 83, 85, 89, 94, 97, 104, 110, 112, 115, 122, 127, 131, 137, 140, 142, 148, 155, 157, 161, 166, 169, 176, 182, 184, 187, 193, 200, 202, 208, 211, 215, 220, 223, 229, 236, 238, 244, 247, 251, 257

Offset: 1

Clark Kimberling, Mar 11 2023

This is the third of four sequences that partition the positive integers. Suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.   Let u' and v' be their (increasing) complements, and consider these four sequences:
(1) u o v, defined by (u o v)(n)  = u(v(n));
(2) u o v';
(3) u' o v;
(4) v' o u'.
Every positive integer is in exactly one of the four sequences. Their limiting densities are 4/9, 2/9, 2/9, 1/9 (and likewise for A360394-A360397 and A360402-A360405).

			(1)  u o v = (5, 8, 10, 12, 15, 16, 18, 21, 24, 26, 27, 30, 31, 35, 37, 39, ...) = A360398
(2)  u o v' = (1, 3, 6, 9, 14, 19, 23, 28, 33, 36, 41, 46, 51, 54, 60, 63, 68, ...) = A360399
(3)  u' o v = (7, 13, 20, 22, 29, 32, 34, 40, 47, 49, 53, 58, 62, 67, 74, 76, ...) = A360400
(4)  u' o v' = (2, 4, 11, 17, 25, 38, 43, 56, 64, 71, 79, 92, 101, 106, 119, ...) = A360401

Cf. A026530, A356133, A360392, A360393, A360398, A286354, A286356, A360394 (intersections instead of results of composition), A360402-A360405.

z = 2000;
u = Accumulate[1 + ThueMorse /@ Range[0, 600]]; (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2;  (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
zz = 100;
Table[u[[v[[n]]]], {n, 1, zz}]    (* A360398 *)
Table[u[[v1[[n]]]], {n, 1, zz}]   (* A360399 *)
Table[u1[[v[[n]]]], {n, 1, zz}]   (* A360400 *)
Table[u1[[v1[[n]]]], {n, 1, zz}]  (* A360401 *)

A010831 Expansion of Product_{k>=1} (1-x^k)^26.

Original entry on oeis.org

1, -26, 299, -1950, 7475, -13754, -12220, 132756, -276575, 0, 1010100, -1486030, -519961, 2486300, 829725, -2215486, -11643060, 18523050, 16317925, -42861650, 0, 11010090, 59644221, -5743400, -138219900

Offset: 0

N. J. A. Sloane

sign

			1 - 26*x + 299*x^2 - 1950*x^3 + 7475*x^4 - 13754*x^5 - 12220*x^6 + 132756*x^7 + ...

Morris 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.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Column k=26 of A286354.

Cf. A000203, A126581, A322433.

CoefficientList[Expand@ Product[(1 - x^k)^26, {k, 25}], x, 25] (* Michael De Vlieger, Jun 08 2016 *)

a(0) = 1, a(n) = -(26/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010839 Expansion of Product_{k >= 1} (1-x^k)^48.

Original entry on oeis.org

1, -48, 1080, -15040, 143820, -985824, 4857920, -16295040, 28412910, 38671600, -424520544, 1268350272, -1211937160, -4306546080, 18293091840, -23522231424, -26299018683, 137218594320, -150999182320, -134713340160

Offset: 0

N. J. A. Sloane

sign

			1 - 48*x + 1080*x^2 - 15040*x^3 + 143820*x^4 - 985824*x^5 + 4857920*x^6 - 16295040*x^7 + ...

Morris 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.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Column k=48 of A286354.

Cf. A000203, A082558, A126581, A282330 (E_8^3), A282332 (E_6*E_8*E_10 = E4*E_10^2), A290009, A290010.

Let b(q) be the determinant of the 3 X 3 Hankel matrix [E_4, E_6, E_8 ; E_6, E_8, E_10 ; E_8, E_10, E_12]. G.f. is -691*b(q)/(q^2*1728^2*250). - Seiichi Manyama, Jul 17 2017
a(n) = (A290010(n+2) - A290009(n+2) + 691*(A282330(n+2) - A282332(n+2)))/(1728^2*250). - Seiichi Manyama, Jul 19 2017
a(0) = 1, a(n) = -(48/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010830 Expansion of Product_{k>=1} (1-x^k)^25.

Original entry on oeis.org

1, -25, 275, -1700, 6050, -9405, -15550, 107525, -182875, -81675, 756655, -801550, -662975, 1220175, 1361350, -209440, -9601900, 8608900, 14889050, -19948500, -6262465, -7057550, 38788925, 19716425, -69119875, 23579969, -82427400, 98068850, 191984400

Offset: 0

N. J. A. Sloane

sign

Morris 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.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Column k=25 of A286354.

Cf. A000203.

a(0) = 1, a(n) = -(25/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010832 Expansion of Product_{k>=1} (1-x^k)^27.

Original entry on oeis.org

1, -27, 324, -2223, 9126, -19278, -5967, 159030, -399087, 151593, 1270971, -2500875, 74970, 4203522, -1004157, -4796037, -11750778, 32885190, 10452375, -77533092, 27104868, 43070625, 63798840, -69960267, -215939061, 236414349, -37046646, 237487433, 85921371

Offset: 0

N. J. A. Sloane

sign

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Column k=27 of A286354.

Cf. A000203.

a(0) = 1, a(n) = -(27/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

cp's OEIS Frontend

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

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A008705 Coefficient of x^n in (Product_{m=1..n}(1-x^m))^n.

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A360401 a(n) = A356133(A360393(n)).

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A010840 Expansion of Product_{k>=1} (1-x^k)^40.

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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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A360400 a(n) = A356133(A360392(n)).

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A010831 Expansion of Product_{k>=1} (1-x^k)^26.

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A010839 Expansion of Product_{k >= 1} (1-x^k)^48.

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