A286354 -id:A286354 - cp's OEIS frontend

A010833 Expansion of Product_{k>=1} (1-x^k)^28.

1, -28, 350, -2520, 11025, -26180, 4158, 184600, -554400, 401100, 1496964, -3920280, 1444625, 6224400, -4972350, -7121296, -8308965, 50796900, -8971200, -121968000, 94011435, 80598288, 20282500, -175228200

Offset: 0

N. J. A. Sloane

sign

			1 - 28*x + 350*x^2 - 2520*x^3 + 11025*x^4 - 26180*x^5 + 4158*x^6 + 184600*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=28 of A286354.

Cf. A000203, A126581.

a(0) = 1, a(n) = -(28/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010834 Expansion of Product_{k>=1} (1-x^k)^29.

Original entry on oeis.org

1, -29, 377, -2842, 13195, -34684, 19285, 206973, -745706, 782275, 1621564, -5803161, 4026360, 8149841, -12056025, -7428263, 254504, 69194580, -49156653, -167517050, 224634319, 94868280, -112333182, -288914501, -172722550, 1061590530, -420678727, -212254364

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=29 of A286354.

Cf. A000203.

a(0) = 1, a(n) = -(29/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010835 Expansion of Product_{k>=1} (1-x^k)^30.

Original entry on oeis.org

1, -30, 405, -3190, 15660, -45036, 40745, 222750, -974835, 1334580, 1547469, -8174520, 8380245, 9200250, -23243355, -2643380, 14704740, 82050570, -116275500, -195804810, 442809990, 25147930, -371898000, -313802910, 125394405, 1688931000, -1364323095, -737497840, 158838945, -1653918750, 6309965146, -1076120370

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=30 of A286354.

Cf. A000203, A082556.

N=66; x='x+O('x^N); /* that many terms */
gf=eta(x)^30;
Vec(gf)  /* show terms */ /* Joerg Arndt, Jul 30 2011 */

a(0) = 1, a(n) = -(30/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010836 Expansion of Product_{k>=1} (1-x^k)^31.

Original entry on oeis.org

1, -31, 434, -3565, 18445, -57505, 70091, 227447, -1241550, 2102730, 1139498, -11000164, 15185009, 8060465, -39266925, 11975548, 33735905, 79961555, -212042635, -176681400, 762467041, -231771190, -762218948, -59474275, 687626655, 2193123086, -3317871844

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=31 of A286354.

Cf. A000203.

a(0) = 1, a(n) = -(31/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010837 Expansion of Product_{k>=1} (1-x^k)^32.

Original entry on oeis.org

1, -32, 464, -3968, 21576, -72384, 109120, 215296, -1542684, 3135712, 217248, -14153856, 25215616, 2704192, -60182656, 43083520, 52111434, 50631680, -328746320, -68928128, 1172526144, -825260672, -1202344640

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=32 of A286354.

Cf. A000203, A082557.

a(0) = 1, a(n) = -(32/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010838 Expansion of Product_{k>=1} (1-x^k)^44.

Original entry on oeis.org

1, -44, 902, -11352, 96965, -582692, 2428382, -6245448, 3684670, 43828180, -195750104, 340202584, 211248851, -2418539816, 4734800950, -43313600, -16560186918, 26632794760, 4021681554, -50231748600, 12519655368

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=44 of A286354.

Cf. A000203.

With[{nn=20},CoefficientList[Series[Product[(1-x^k)^44,{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Mar 23 2015 *)

a(0) = 1, a(n) = -(44/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

A010841 Expansion of Product_{k>=1} (1-x^k)^64.

Original entry on oeis.org

1, -64, 1952, -37632, 512400, -5207936, 40618368, -244952576, 1124362248, -3684692800, 6607738816, 8603838208, -109557823168, 389162471040, -599467398400, -815811136000, 6834665221028, -15689583552384, 5284986829472, 66706108652800, -183175485196256, 124242038746624

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=64 of A286354.

Cf. A000203, A082559.

nmax=20; CoefficientList[Series[Product[(1-x^k)^64,{k,nmax}],{x,0,nmax}],x] (* Stefano Spezia, May 27 2025 *)

a(0) = 1, a(n) = -(64/n) * Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Aug 13 2023

cp's OEIS Frontend

A010833 Expansion of Product_{k>=1} (1-x^k)^28.

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A010834 Expansion of Product_{k>=1} (1-x^k)^29.

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A010835 Expansion of Product_{k>=1} (1-x^k)^30.

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A010836 Expansion of Product_{k>=1} (1-x^k)^31.

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A010837 Expansion of Product_{k>=1} (1-x^k)^32.

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A010838 Expansion of Product_{k>=1} (1-x^k)^44.

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A010841 Expansion of Product_{k>=1} (1-x^k)^64.

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