A126581 -id:A126581 - cp's OEIS frontend

A003038 Dimensions of split simple Lie algebras over any field of characteristic zero.

3, 8, 10, 14, 15, 21, 24, 28, 35, 36, 45, 48, 52, 55, 63, 66, 78, 80, 91, 99, 105, 120, 133, 136, 143, 153, 168, 171, 190, 195, 210, 224, 231, 248, 253, 255, 276, 288, 300, 323, 325, 351, 360, 378, 399, 406, 435, 440, 465, 483, 496, 528, 561, 575, 595, 624, 630

Offset: 1

N. J. A. Sloane

			The Lie algebras in question and their dimensions are the following:
A_l: l(l+2), l >= 1,
B_l: l(2l+1), l >= 2,
C_l: l(2l+1), l >= 3,
D_l: l(2l-1), l >= 4,
G_2: 14, F_4: 52, E_6: 78, E_7: 133, E_8: 248.

Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
N. Jacobson, Lie Algebras. Wiley, NY, 1962; pp. 141-146.
I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal., 13 (1982), 988-1007.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A001066, A126581.

Subsequences, apart from some initial terms: A000217, A000384, A005563, A014105.

import Data.Set (deleteFindMin, fromList, insert)
a003038 n = a003038_list !! (n-1)
a003038_list = f (fromList (3 : [14, 52, 78, 133, 248]))
   (drop 2 a005563_list) (drop 4 a000217_list) where
   f s (x:xs) (y:ys) = m : f (x `insert` (y `insert` s')) xs ys where
     (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Dec 16 2012

M:=4200; M2:=M^2; sa:=[seq(l*(l+2),l=1..M)]; sb:=[seq(l*(2*l+1),l=2..M)]; sd:=[seq(l*(2*l-1),l=4..M)]; se:=[14,52,78,133,248]; s:=convert(sa,set) union convert(sb,set) union convert(sd,set) union convert(se,set); t:=convert(s,list); for i from 1 to nops(t) do if t[i] <= M2 then lprint(i,t[i]); fi; od:

max = 26; sa = Table[ k*(k+2), {k, 1, max}]; sb = Table[ k*(2k+1), {k, 2, max}]; sd:= Table[ k*(2k-1), {k, 4, max}]; se = {14, 52, 78, 133, 248}; Select[ Union[sa, sb, sd, se], # <= max^2 &](* Jean-François Alcover, Nov 18 2011, after Maple *)

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004

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

A010827 Expansion of Product_{k>=1} (1 - x^k)^21.

Original entry on oeis.org

1, -21, 189, -910, 2205, -378, -13321, 33345, -10395, -86870, 122703, 46683, -98287, -264915, 96390, 1163064, -1113588, -1066527, 1042055, 536025, 2287467, -3603805, -1391733, 478170, -562555, 13742379, -7889805

Offset: 0

N. J. A. Sloane

sign

If not n == 0 (mod 7) then a(n) == 0 (mod 7). The reverse is not true, e.g., a(14) = 96390 == (0 mod 7). See the Hardy reference, p. 165. - Wolfdieter Lang, Jan 28 2017

			G.f. = 1 - 21*x + 189*x^2 - 910*x^3 + 2205*x^4 - 378*x^5 - 13321*x^6 + 33345*x^7 + ...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 165.
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. MR1955423 (2003k:11071).
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A126581.

CoefficientList[Expand@ Product[(1 - x^k)^21, {k, 27}], x, 27] (* Michael De Vlieger, Jun 08 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^21, {x, 0, n}]; (* Michael Somos, Jan 28 2017 *)

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^21, n))}; /* Michael Somos, Jan 28 2017 */

G.f.: Product_{k>0} (1 - x^k)^21.
a(0) = 1, a(n) = -(21/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(-21*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

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

Original entry on oeis.org

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

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A003038 Dimensions of split simple Lie algebras over any field of characteristic zero.

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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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A010827 Expansion of Product_{k>=1} (1 - x^k)^21.

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

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