A292929 G.f.: A(x,q) = sqrt( Q(x,q) / Q(x,-q) ), where Q(x,q) = Sum_{n=-oo..+oo} (x - q^n)^n.

1, 2, -2, 2, -4, 2, 2, -4, 6, -4, 2, -4, 8, -12, 6, 2, -4, 8, -14, 16, -8, 2, -4, 8, -12, 18, -24, 12, 2, -4, 8, -12, 20, -36, 38, -16, 2, -4, 8, -12, 24, -44, 56, -52, 22, 2, -4, 8, -12, 24, -40, 52, -74, 74, -30, 2, -4, 8, -12, 24, -32, 38, -76, 116, -104, 40, 2, -4, 8, -12, 24, -32, 48, -96, 136, -164, 142, -52, 2, -4, 8, -12, 24, -32, 64, -124, 138, -164, 224, -192, 68, 2, -4, 8, -12, 24, -32, 64, -100, 86, -134, 252, -324, 258, -88, 2, -4, 8, -12, 24, -32, 64, -68, 32, -148, 316, -396, 442, -340, 112, 2, -4, 8, -12, 24, -32, 64, -68, 88, -276, 398, -384, 482, -592, 446, -144, 2, -4, 8, -12, 24, -32, 64, -68, 152, -376, 328, -192, 384, -684, 808, -584, 182, 2, -4, 8, -12, 24, -32, 64, -68, 152, -248, 24, -22, 462, -790, 990, -1074, 752, -228, 2, -4, 8, -12, 24, -32, 64, -68, 152, -120, -152, -288, 1048, -1064, 982, -1272, 1410, -964, 286, 2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 136, -988, 1402, -708, 548, -1168, 1748, -1860, 1232, -356

Offset

0,2

Comments

Compare to the g.f. of A108494: sqrt( theta_4(q) / theta_4(-q) ).

Note the related identities:

(1) Sum_{n=-oo..+oo} (x - q^n)^(n-1) = 0.

(2) Sum_{n=-oo..+oo} (x - q^n)^(n+1) = x * Sum_{n=-oo..+oo} (x - q^n)^n.

(3) Sum_{n=-oo..+oo} (x - q^n)^n = 1/(1-x) + Sum_{n>=1} (-1)^n * q^(n^2) * (2 - x*q^n)/(1 - x*q^n)^(n+1).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..5150 as read by antidiagonals 0..100 of a rectangular array.

Formula

Antidiagonal sums equal zero after the initial '1'.

G.f. of Row 0: Product_{n>=1} (1 - q^(2*n-1)) / (1 + q^(2*n-1)); see A108494.

G.f. of Row 1: 2*q * Product_{n>=1} (1 + q^(2*n))/((1 + q^n)*(1 + q^(2*n-1))*(1 + q^(4*n))).

Example

G.f.: A(x,q) = Sum_{n>=0} x^n * Sum_{k>=0} T(n,k) * q^(n+k), where
A(x,q) = sqrt( Q(x,q) / Q(x,-q) ) and Q(x,q) is the g.f. of A293600:
Q(x,q) = (1 - 2*q + 2*q^4 - 2*q^9 + 2*q^16 - 2*q^25 + 2*q^36 +...)
+ x*(1 - 3*q^2 + 5*q^6 - 7*q^12 + 9*q^20 - 11*q^30 + 13*q^42 +...)
+ x^2*(1 - 4*q^3 + 9*q^8 - 16*q^15 + 25*q^24 - 36*q^35 + 49*q^48 +...)
+ x^3*(1 - 5*q^4 + 14*q^10 - 30*q^18 + 55*q^28 - 91*q^40 + 140*q^54 +...)
+ x^4*(1 - 6*q^5 + 20*q^12 - 50*q^21 + 105*q^32 - 196*q^45 + 336*q^60 +...)
+ x^5*(1 - 7*q^6 + 27*q^14 - 77*q^24 + 182*q^36 - 378*q^50 + 714*q^66 +...)
+ x^6*(1 - 8*q^7 + 35*q^16 - 112*q^27 + 294*q^40 - 672*q^55 + 1386*q^72 +...)
+ x^7*(1 - 9*q^8 + 44*q^18 - 156*q^30 + 450*q^44 - 1122*q^60 + 792*q^78 +...)
+ ...
Explicitly, the g.f. of this table begins:
A(x,q) = (1 - 2*q + 2*q^2 - 4*q^3 + 6*q^4 - 8*q^5 + 12*q^6 - 16*q^7 + 22*q^8 - 30*q^9 + 40*q^10 - 52*q^11 + 68*q^12 - 88*q^13 +...)
+ x*(2*q - 4*q^2 + 6*q^3 - 12*q^4 + 16*q^5 - 24*q^6 + 38*q^7 - 52*q^8 + 74*q^9 - 104*q^10 + 142*q^11 - 192*q^12 + 258*q^13 - 340*q^14 +...)
+ x^2*(2*q^2 - 4*q^3 + 8*q^4 - 14*q^5 + 18*q^6 - 36*q^7 + 56*q^8 - 74*q^9 + 116*q^10 - 164*q^11 + 224*q^12 - 324*q^13 + 442*q^14 - 592*q^15 +...)
+ x^3*(2*q^3 - 4*q^4 + 8*q^5 - 12*q^6 + 20*q^7 - 44*q^8 + 52*q^9 - 76*q^10 + 136*q^11 - 164*q^12 + 252*q^13 - 396*q^14 + 482*q^15 - 684*q^16 +...)
+ x^4*(2*q^4 - 4*q^5 + 8*q^6 - 12*q^7 + 24*q^8 - 40*q^9 + 38*q^10 - 96*q^11 + 138*q^12 - 134*q^13 + 316*q^14 - 384*q^15 + 384*q^16 - 790*q^17 +...)
+ x^5*(2*q^5 - 4*q^6 + 8*q^7 - 12*q^8 + 24*q^9 - 32*q^10 + 48*q^11 - 124*q^12 + 86*q^13 - 148*q^14 + 398*q^15 - 192*q^16 + 462*q^17 - 1064*q^18 +...)
+ x^6*(2*q^6 - 4*q^7 + 8*q^8 - 12*q^9 + 24*q^10 - 32*q^11 + 64*q^12 - 100*q^13 + 32*q^14 - 276*q^15 + 328*q^16 - 22*q^17 + 1048*q^18 - 708*q^19 +...)
+ x^7*(2*q^7 - 4*q^8 + 8*q^9 - 12*q^10 + 24*q^11 - 32*q^12 + 64*q^13 - 68*q^14 + 88*q^15 - 376*q^16 + 24*q^17 - 288*q^18 + 1402*q^19 + 936*q^20 +...)
+ x^8*(2*q^8 - 4*q^9 + 8*q^10 - 12*q^11 + 24*q^12 - 32*q^13 + 64*q^14 - 68*q^15 + 152*q^16 - 248*q^17 - 152*q^18 - 988*q^19 + 554*q^20 + 1554*q^21 +...)
+ x^9*(2*q^9 - 4*q^10 + 8*q^11 - 12*q^12 + 24*q^13 - 32*q^14 + 64*q^15 - 68*q^16 + 152*q^17 - 120*q^18 + 136*q^19 - 1276*q^20 - 1016*q^21 - 912*q^22+...)
+ x^10*(2*q^10 - 4*q^11 + 8*q^12 - 12*q^13 + 24*q^14 - 32*q^15 + 64*q^16 - 68*q^17 + 152*q^18 - 120*q^19 + 392*q^20 - 636*q^21 - 1432*q^22 - 4352*q^23 +...)
+ x^11*(2*q^11 - 4*q^12 + 8*q^13 - 12*q^14 + 24*q^15 - 32*q^16 + 64*q^17 - 68*q^18 + 152*q^19 - 120*q^20 + 392*q^21 - 124*q^22 - 24*q^23 - 4800*q^24+...)
+ x^12*(2*q^12 - 4*q^13 + 8*q^14 - 12*q^15 + 24*q^16 - 32*q^17 + 64*q^18 - 68*q^19 + 152*q^20 - 120*q^21 + 392*q^22 - 124*q^23 + 1000*q^24 - 1728*q^25 +...)
+ ...
G.F. OF ROWS.
The coefficient of x^0 in A(x,q) is
(R0) Product_{n>=1} (1 - q^(2*n-1)) / (1 + q^(2*n-1)).
The coefficient of x in A(x,q) is
(R1) 2*q * Product_{n>=1} (1 + q^(2*n))/((1 + q^n)*(1 + q^(2*n-1))*(1 + q^(4*n))).
RECTANGULAR ARRAY.
This table of coefficients T(n,k) of x^n*y^(n+k) in A(x,q) begins:
[1, -2, 2, -4, 6, -8, 12, -16, 22, -30, 40, -52, 68, -88, 112, -144, ...];
[2, -4, 6, -12, 16, -24, 38, -52, 74, -104, 142, -192, 258, -340, 446, ...];
[2, -4, 8, -14, 18, -36, 56, -74, 116, -164, 224, -324, 442, -592, 808, ...];
[2, -4, 8, -12, 20, -44, 52, -76, 136, -164, 252, -396, 482, -684, 990, ...];
[2, -4, 8, -12, 24, -40, 38, -96, 138, -134, 316, -384, 384, -790, 982, ...];
[2, -4, 8, -12, 24, -32, 48, -124, 86, -148, 398, -192, 462, -1064, 548, ...];
[2, -4, 8, -12, 24, -32, 64, -100, 32, -276, 328, -22, 1048, -708, -220, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 88, -376, 24, -288, 1402, 936, 1146, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -248, -152, -988, 554, 1554, 5628, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 136, -1276, -1016, -912, 6428, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -636, -1432, -4352, -320, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, -24, -4800, -7696, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, -1728, -7696, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, -1040, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, 2836, ...]; ...
The limit of the rows approach A293601, which begins:
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, 2836, 10280, 15112, 38668, 68348, 154152, 297948, 633352, 1269884, 2649892, ...].
RATIOS OF ROW G.F.
The ratios of the row generating functions are as follows.
2 + 2*q^2 + 2*q^6 + 2*q^8 + 2*q^10 + 2*q^12 + 2*q^14 +...
1 + q^2 + q^3 - 3*q^5 + q^6 + 4*q^7 + q^8 - 3*q^9 + q^10 + 3*q^11 +...
1 + q^3 + 3*q^4 - 2*q^5 - 11*q^6 - 3*q^7 + 25*q^8 + 29*q^9 - 33*q^10 +...
1 + 2*q^4 + 6*q^5 - 3*q^6 - 28*q^7 - 27*q^8 + 39*q^9 + 160*q^10 +...
1 + 4*q^5 + 13*q^6 - 4*q^7 - 62*q^8 - 85*q^9 + 19*q^10 + 334*q^11 +...
1 + 8*q^6 + 28*q^7 - 3*q^8 - 134*q^9 - 219*q^10 - 43*q^11 + 571*q^12 +...
1 + 16*q^7 + 60*q^8 + 6*q^9 - 284*q^10 - 557*q^11 - 229*q^12 + 1264*q^13 +...
1 + 32*q^8 + 128*q^9 + 40*q^10 - 590*q^11 - 1380*q^12 - 875*q^13 +...
1 + 64*q^9 + 272*q^10 + 144*q^11 - 1201*q^12 - 3347*q^13 - 2866*q^14 +...
1 + 128*q^10 + 576*q^11 + 432*q^12 - 2392*q^13 - 7966*q^14 - 8598*q^15 +...
1 + 256*q^11 + 1216*q^12 + 1184*q^13 - 4648*q^14 - 18642*q^15 +...
...

Crossrefs

Cf. A293600, A293601, A108494 (row 0), A293132 (row 1), A294065 (row 2), A294066 (row 3), A294067 (row 4).

Sequence in context: A159802 A329586 A255336 * A049627 A278223 A368556

Adjacent sequences: A292926 A292927 A292928 * A292930 A292931 A292932

Keyword

sign,tabl

Author

Paul D. Hanna, Oct 22 2017

Status

approved