A181101 -id:A181101 - cp's OEIS frontend

A181105 Triangle T(n,k) read by rows. T(n,k)=abs(A181101)*n/k.

1, 2, 0, 3, 0, 0, 0, 2, 0, 0, 5, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Gary W. Adamson, Oct 03 2010

Let A=T(n,k)*T(n,k), B=A*A, C=B*B, D=C*C, and so on. Then the first column in the last matrix (D) converges to the natural numbers. Rows sums are (unsurprisingly) A020639. Except for the first column, the largest and last element in the columns are also A020639.

			Triangle starts:
1,
2,0,
3,0,0,
0,2,0,0,
5,0,0,0,0,
0,0,2,0,0,0,
7,0,0,0,0,0,0,
0,0,0,2,0,0,0,0,
0,0,3,0,0,0,0,0,0,
0,0,0,0,2,0,0,0,0,0,
11,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,2,0,0,0,0,0,0,
13,0,0,0,0,0,0,0,0,0,0,0,0,

Cf. A181101, A020639.

A030209 Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.

Original entry on oeis.org

1, -2, -3, 4, 6, 6, -16, -8, 9, -12, 12, -12, 38, 32, -18, 16, -126, -18, 20, 24, 48, -24, 168, 24, -89, -76, -27, -64, 30, 36, -88, -32, -36, 252, -96, 36, 254, -40, -114, -48, 42, -96, -52, 48, 54, -336, -96, -48, -87, 178, 378, 152, 198, 54, 72, 128

Offset: 1

N. J. A. Sloane

Identical to table 1, p. 493, of Alaca citation. - Jonathan Vos Post, May 24 2007
Unique cusp form of weight 4 for congruence group Gamma_1(6). - Michael Somos, Aug 11 2011
Number 14 of the 74 eta-quotients listed in Table I of Martin (1996).
The table 1, p. 493 of Alaca reference is the first 50 values of c_6(n). - Michael Somos, May 17 2015

			G.f. = q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^5 + 6*q^6 - 16*q^7 - 8*q^8 + 9*q^9 - 12*q^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Saban Alaca and Kenneth Williams, Evaluation of the convolution sums sum_{l+6m=n} sigma(l)*sigma(m) and sum_{2l+3m=n} sigma(l)*sigma(m), J. Number Theory 124 (2007), no. 2, 491-510. MR2321376 (2008a:11005)
K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA, 104 (2010), 3725-3731.
Brian Conrey and David Farmer (?), Eta Products and Quotients which are Newforms.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
LMFDB, Space of modular forms of level 6 and weight 4.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A030188, A181101, A186100.

Basis( CuspForms( Gamma1(6), 4), 57) [1]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3] QPochhammer[ q^6])^2, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A))^2, n))}; /* Michael Somos, Feb 14 2006 */

CuspForms( Gamma1(6), 4, prec = 57).0; # Michael Somos, Aug 11 2011

Euler transform of period 6 sequence [ -2, -4, -4, -4, -2, -8, ...]. - Michael Somos, Feb 13 2006
a(n) is multiplicative with a(p^e) = (-p)^e if p<5, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 36 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 11 2011
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)))^2.
a(2*n) = -2 * a(n). Convolution square of A030188. - Michael Somos, May 27 2012
Convolution with A181102 is A186100. - Michael Somos, Jul 07 2015

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A181105 Triangle T(n,k) read by rows. T(n,k)=abs(A181101)*n/k.

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A030209 Expansion of (eta(q) * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q.

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