This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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: A066671 A159802 A255336 * A049627 A278223 A134058 Adjacent sequences:  A292926 A292927 A292928 * A292930 A292931 A292932 KEYWORD sign,tabl AUTHOR Paul D. Hanna, Oct 22 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)