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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. 7

%I

%S 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,

%T -24,12,2,-4,8,-12,20,-36,38,-16,2,-4,8,-12,24,-44,56,-52,22,2,-4,8,

%U -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

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

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

%C Note the related identities:

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

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

%C (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).

%H Paul D. Hanna, <a href="/A292929/b292929.txt">Table of n, a(n) for n = 0..5150 as read by antidiagonals 0..100 of a rectangular array.</a>

%F Antidiagonal sums equal zero after the initial '1'.

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

%F 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))).

%e G.f.: A(x,q) = Sum_{n>=0} x^n * Sum_{k>=0} T(n,k) * q^(n+k), where

%e A(x,q) = sqrt( Q(x,q) / Q(x,-q) ) and Q(x,q) is the g.f. of A293600:

%e Q(x,q) = (1 - 2*q + 2*q^4 - 2*q^9 + 2*q^16 - 2*q^25 + 2*q^36 +...)

%e + x*(1 - 3*q^2 + 5*q^6 - 7*q^12 + 9*q^20 - 11*q^30 + 13*q^42 +...)

%e + x^2*(1 - 4*q^3 + 9*q^8 - 16*q^15 + 25*q^24 - 36*q^35 + 49*q^48 +...)

%e + x^3*(1 - 5*q^4 + 14*q^10 - 30*q^18 + 55*q^28 - 91*q^40 + 140*q^54 +...)

%e + x^4*(1 - 6*q^5 + 20*q^12 - 50*q^21 + 105*q^32 - 196*q^45 + 336*q^60 +...)

%e + x^5*(1 - 7*q^6 + 27*q^14 - 77*q^24 + 182*q^36 - 378*q^50 + 714*q^66 +...)

%e + x^6*(1 - 8*q^7 + 35*q^16 - 112*q^27 + 294*q^40 - 672*q^55 + 1386*q^72 +...)

%e + x^7*(1 - 9*q^8 + 44*q^18 - 156*q^30 + 450*q^44 - 1122*q^60 + 792*q^78 +...)

%e + ...

%e Explicitly, the g.f. of this table begins:

%e 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 +...)

%e + 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 +...)

%e + 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 +...)

%e + 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 +...)

%e + 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 +...)

%e + 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 +...)

%e + 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 +...)

%e + 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 +...)

%e + 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 +...)

%e + 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+...)

%e + 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 +...)

%e + 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+...)

%e + 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 +...)

%e + ...

%e G.F. OF ROWS.

%e The coefficient of x^0 in A(x,q) is

%e (R0) Product_{n>=1} (1 - q^(2*n-1)) / (1 + q^(2*n-1)).

%e The coefficient of x in A(x,q) is

%e (R1) 2*q * Product_{n>=1} (1 + q^(2*n))/((1 + q^n)*(1 + q^(2*n-1))*(1 + q^(4*n))).

%e RECTANGULAR ARRAY.

%e This table of coefficients T(n,k) of x^n*y^(n+k) in A(x,q) begins:

%e [1, -2, 2, -4, 6, -8, 12, -16, 22, -30, 40, -52, 68, -88, 112, -144, ...];

%e [2, -4, 6, -12, 16, -24, 38, -52, 74, -104, 142, -192, 258, -340, 446, ...];

%e [2, -4, 8, -14, 18, -36, 56, -74, 116, -164, 224, -324, 442, -592, 808, ...];

%e [2, -4, 8, -12, 20, -44, 52, -76, 136, -164, 252, -396, 482, -684, 990, ...];

%e [2, -4, 8, -12, 24, -40, 38, -96, 138, -134, 316, -384, 384, -790, 982, ...];

%e [2, -4, 8, -12, 24, -32, 48, -124, 86, -148, 398, -192, 462, -1064, 548, ...];

%e [2, -4, 8, -12, 24, -32, 64, -100, 32, -276, 328, -22, 1048, -708, -220, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 88, -376, 24, -288, 1402, 936, 1146, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -248, -152, -988, 554, 1554, 5628, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 136, -1276, -1016, -912, 6428, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -636, -1432, -4352, -320, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, -24, -4800, -7696, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, -1728, -7696, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, -1040, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, ...];

%e [2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, 2836, ...]; ...

%e The limit of the rows approach A293601, which begins:

%e [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, ...].

%e RATIOS OF ROW G.F.

%e The ratios of the row generating functions are as follows.

%e 2 + 2*q^2 + 2*q^6 + 2*q^8 + 2*q^10 + 2*q^12 + 2*q^14 +...

%e 1 + q^2 + q^3 - 3*q^5 + q^6 + 4*q^7 + q^8 - 3*q^9 + q^10 + 3*q^11 +...

%e 1 + q^3 + 3*q^4 - 2*q^5 - 11*q^6 - 3*q^7 + 25*q^8 + 29*q^9 - 33*q^10 +...

%e 1 + 2*q^4 + 6*q^5 - 3*q^6 - 28*q^7 - 27*q^8 + 39*q^9 + 160*q^10 +...

%e 1 + 4*q^5 + 13*q^6 - 4*q^7 - 62*q^8 - 85*q^9 + 19*q^10 + 334*q^11 +...

%e 1 + 8*q^6 + 28*q^7 - 3*q^8 - 134*q^9 - 219*q^10 - 43*q^11 + 571*q^12 +...

%e 1 + 16*q^7 + 60*q^8 + 6*q^9 - 284*q^10 - 557*q^11 - 229*q^12 + 1264*q^13 +...

%e 1 + 32*q^8 + 128*q^9 + 40*q^10 - 590*q^11 - 1380*q^12 - 875*q^13 +...

%e 1 + 64*q^9 + 272*q^10 + 144*q^11 - 1201*q^12 - 3347*q^13 - 2866*q^14 +...

%e 1 + 128*q^10 + 576*q^11 + 432*q^12 - 2392*q^13 - 7966*q^14 - 8598*q^15 +...

%e 1 + 256*q^11 + 1216*q^12 + 1184*q^13 - 4648*q^14 - 18642*q^15 +...

%e ...

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

%K sign,tabl

%O 0,2

%A _Paul D. Hanna_, Oct 22 2017

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Last modified January 25 23:04 EST 2020. Contains 331270 sequences. (Running on oeis4.)