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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Oct 22 2017

Keywords

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).

Examples

			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 +...
...

Links

Crossrefs

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

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))).

A294065 Row 2 in rectangular array A292929.

Original entry on oeis.org

2, -4, 8, -14, 18, -36, 56, -74, 116, -164, 224, -324, 442, -592, 808, -1074, 1410, -1860, 2416, -3102, 4010, -5112, 6464, -8204, 10294, -12860, 16072, -19914, 24586, -30356, 37248, -45534, 55608, -67604, 81928, -99182, 119608, -143832, 172760, -206834, 247048, -294676, 350504, -416080, 493248, -583340, 688616, -811740, 954974, -1121564, 1315504, -1540210, 1800434, -2102060, 2450224, -2852040
Offset: 0

Views

Author

Paul D. Hanna, Oct 23 2017

Keywords

Examples

			G.f.: A(q) = 2 - 4*q + 8*q^2 - 14*q^3 + 18*q^4 - 36*q^5 + 56*q^6 - 74*q^7 + 116*q^8 - 164*q^9 + 224*q^10 - 324*q^11 + 442*q^12 - 592*q^13 + 808*q^14 - 1074*q^15 + 1410*q^16 - 1860*q^17 + 2416*q^18 - 3102*q^19 + 4010*q^20 +...
RELATED SERIES.
Let R1(q) denote the g.f. of row 1 (with offset 0) in array A292929, then
A(q)/R1(q) = 1 + q^2 + q^3 - 3*q^5 + q^6 + 4*q^7 + q^8 - 3*q^9 + q^10 + 3*q^11 + q^12 - 5*q^13 + q^14 + 7*q^15 - 11*q^17 + 16*q^19 + 2*q^20 - 18*q^21 + 21*q^23 + q^24 - 27*q^25 + q^26 + 38*q^27 + q^28 - 55*q^29 + 2*q^30 +...
then it appears that the even bisection of A(q)/R1(q) forms a g.f. of A053692:
(A(q)/R1(q) + A(-q)/R1(-q))/2 = Product_{n>=1} (1 - q^(16*n))^2*(1 + q^(4*n-2)).

Links

Crossrefs

Cf. A292929, A293132, A294066, A294067.

Programs

  • Mathematica
    nmax = 55; kmax = Ceiling[Sqrt[nmax]];
Q[q_] := Sum[(x - q^k)^k, {k, -kmax, kmax}];
S[q_] := Sqrt[Q[q]/Q[-q]];
row[n_] := (1/q^n)*SeriesCoefficient[Sqrt[Q[q]/Q[-q]], {x, 0, n}] + O[q]^nmax // CoefficientList[#, q]&;
row[2] (* Jean-François Alcover, Nov 04 2017 *)

A294066 Row 3 in rectangular array A292929.

Original entry on oeis.org

2, -4, 8, -12, 20, -44, 52, -76, 136, -164, 252, -396, 482, -684, 990, -1272, 1748, -2388, 3038, -4020, 5358, -6796, 8820, -11448, 14334, -18304, 23320, -28940, 36444, -45708, 56340, -70056, 86698, -106056, 130400, -159852, 194166, -236452, 287272, -346544, 418746, -504800, 604946, -725756, 868892, -1035456, 1234410, -1468436, 1740602, -2063076, 2440838, -2879056, 3394228, -3995400, 4690976
Offset: 0

Views

Author

Paul D. Hanna, Oct 23 2017

Keywords

Examples

			G.f.: A(q) = 2 - 4*q + 8*q^2 - 12*q^3 + 20*q^4 - 44*q^5 + 52*q^6 - 76*q^7 + 136*q^8 - 164*q^9 + 252*q^10 - 396*q^11 + 482*q^12 - 684*q^13 + 990*q^14 - 1272*q^15 + 1748*q^16 - 2388*q^17 + 3038*q^18 - 4020*q^19 + 5358*q^20 +...

Links

Crossrefs

Cf. A292929, A293132, A294065, A294067.

Programs

  • Mathematica
    nmax = 55; kmax = Ceiling[Sqrt[nmax]]+1;
Q[q_] := Sum[(x - q^k)^k, {k, -kmax, kmax}];
S[q_] := Sqrt[Q[q]/Q[-q]];
row[n_] := (1/q^n)*SeriesCoefficient[Sqrt[Q[q]/Q[-q]], {x, 0, n} ] + O[q]^nmax // CoefficientList[#, q]&;
row[3] (* Jean-François Alcover, Nov 04 2017 *)

A294067 Row 4 in rectangular array A292929.

Original entry on oeis.org

2, -4, 8, -12, 24, -40, 38, -96, 138, -134, 316, -384, 384, -790, 982, -1168, 1976, -2400, 2904, -4464, 5632, -6956, 9904, -12320, 15186, -20938, 26000, -32008, 42560, -52278, 64458, -83736, 102294, -125428, 159288, -193908, 236632, -295612, 358170, -434364, 535958, -646032, 778504, -950552, 1139784, -1367002, 1654268, -1972508, 2353214, -2825722, 3355344, -3983820, 4749672, -5614558, 6634830
Offset: 0

Views

Author

Paul D. Hanna, Oct 23 2017

Keywords

Examples

			G.f.: A(q) = 2 - 4*q + 8*q^2 - 12*q^3 + 24*q^4 - 40*q^5 + 38*q^6 - 96*q^7 + 138*q^8 - 134*q^9 + 316*q^10 - 384*q^11 + 384*q^12 - 790*q^13 + 982*q^14 - 1168*q^15 + 1976*q^16 - 2400*q^17 + 2904*q^18 - 4464*q^19 + 5632*q^20 +...

Links

Crossrefs

Cf. A292929, A293132, A294065, A294066.

Programs

  • Mathematica
    nmax = 55; kmax = Ceiling[Sqrt[nmax]]+1;
Q[q_] := Sum[(x - q^k)^k, {k, -kmax, kmax}];
S[q_] := Sqrt[Q[q]/Q[-q]];
row[n_] := (1/q^n)*SeriesCoefficient[Sqrt[Q[q]/Q[-q]], {x, 0, n} ] + O[q]^nmax // CoefficientList[#, q] &;
row[4] (* Jean-François Alcover, Nov 04 2017 *)
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