cp's OEIS Frontend

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.

Showing 1-6 of 6 results.

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 *)

A293601 Limit of rows in rectangular array A292929.

Original entry on oeis.org

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, 5395272, 11157512, 22890976, 47251564, 97224304, 200605456, 413622556, 853809232, 1762332664, 3640315888, 7521114700, 15545862696, 32142131064, 66481012488, 137544496052
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2017

Keywords

Comments

The g.f. of A292929 is R(x,q) = sqrt( Q(x,q) / Q(x,-q) ), where Q(x,q) = Sum_{n=-oo..+oo} (x - q^n)^n; the g.f. of this sequence equals the limit of the coefficient of x^n in R(x,q) / q^n as a power series in q.
a(n+1)/a(n) tends to 2.0946... - Vaclav Kotesovec, Oct 23 2017

Links

Crossrefs

Cf. A292929.

Formula

G.f.: A(q) = 2 - 4*q + 8*q^2 - 12*q^3 + 24*q^4 - 32*q^5 + 64*q^6 - 68*q^7 + 152*q^8 - 120*q^9 + 392*q^10 - 124*q^11 + 1000*q^12 + 320*q^13 + 3056*q^14 + 2836*q^15 + 10280*q^16 + 15112*q^17 + 38668*q^18 +...
Let R(x,q) be the g.f. of A292929, then we can illustrate the g.f. of this sequence as follows.
The coefficient of x^4 in R(x,q) begins:
2*q^4 - 4*q^5 + 8*q^6 - 12*q^7 + 24*q^8 - 40*q^9 + 38*q^10 +...
The coefficient of x^5 in R(x,q) begins:
2*q^5 - 4*q^6 + 8*q^7 - 12*q^8 + 24*q^9 - 32*q^10 + 48*q^11 +...
The coefficient of x^6 in R(x,q) begins:
2*q^6 - 4*q^7 + 8*q^8 - 12*q^9 + 24*q^10 - 32*q^11 + 64*q^12 +...
The g.f. A(q) equals the limit of the coefficient if x^n in R(x,q)/q^n.

A293132 G.f.: 2*q * Product_{n>=1} (1 + q^(2*n))/((1 + q^n)*(1 + q^(2*n-1))*(1 + q^(4*n))) in powers of q.

Original entry on oeis.org

2, -4, 6, -12, 16, -24, 38, -52, 74, -104, 142, -192, 258, -340, 446, -584, 756, -972, 1244, -1580, 1996, -2516, 3148, -3924, 4878, -6032, 7434, -9136, 11182, -13644, 16608, -20148, 24378, -29428, 35422, -42540, 50978, -60940, 72700, -86556, 102838, -121952, 144360, -170564, 201176, -236900, 278494, -326876, 383094, -448288, 523824, -611248, 712256, -828860, 963324, -1118160, 1296296, -1501028, 1736030, -2005540
Offset: 1

Views

Author

Paul D. Hanna, Oct 22 2017

Keywords

Comments

G.f. of row 1 in rectangular array A292929.

Examples

			G.f.: A(q) = 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 +...

Links

Crossrefs

Cf. A292929, A294065, A294066, A294067.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[2*Product[1/((1 + x^(2*k-1))^2 * (1 + x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 23 2017 *)
  • PARI
    {a(n) = polcoeff( 2*q * prod(m=1,n, (1 + q^(2*m))/((1 + q^m)*(1 + q^(2*m-1))*(1 + q^(4*m)) +q*O(q^n))),n,q)}
for(n=1,60,print1(a(n),", "))

Formula

a(n) ~ -(-1)^n * 7^(1/4) * exp(sqrt(7*n/3)*Pi/2) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 23 2017

A293600 G.f. A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1), as a flattened rectangular array of coefficients T(n,k) of x^n * y^(k*(n+k-1)) in A(x,y) for n>=1.

Original entry on oeis.org

1, 1, -2, 1, -3, 2, 1, -4, 5, -2, 1, -5, 9, -7, 2, 1, -6, 14, -16, 9, -2, 1, -7, 20, -30, 25, -11, 2, 1, -8, 27, -50, 55, -36, 13, -2, 1, -9, 35, -77, 105, -91, 49, -15, 2, 1, -10, 44, -112, 182, -196, 140, -64, 17, -2, 1, -11, 54, -156, 294, -378, 336, -204, 81, -19, 2, 1, -12, 65, -210, 450, -672, 714, -540, 285, -100, 21, -2, 1, -13, 77, -275, 660, -1122, 1386, -1254, 825, -385, 121, -23, 2, 1, -14, 90, -352, 935, -1782, 2508, -2640, 2079, -1210, 506, -144, 25, -2, 1, -15, 104, -442, 1287, -2717, 4290, -5148, 4719, -3289, 1716, -650, 169, -27, 2
Offset: 1

Views

Author

Paul D. Hanna, Oct 16 2017

Keywords

Comments

Compare g.f. to the identity: Sum_{-oo..+oo} (x - y^n)^(n-1) = 0.
The Lucas triangle, A029635, consists of essentially the same coefficients, but differs in signs and initial term.

Examples

			G.f. A(x,y) = Sum_{n>=1} x^n * Sum_{k>=0} T(n,k) * y^(k*(n+k-1))
such that A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1).
Explicitly, the g.f. of this array begins:
A(x,y) = x*(1 - 2*y + 2*y^4 - 2*y^9 + 2*y^16 - 2*y^25 + 2*y^36 +...)
+ x^2*(1 - 3*y^2 + 5*y^6 - 7*y^12 + 9*y^20 - 11*y^30 + 13*y^42 +...)
+ x^3*(1 - 4*y^3 + 9*y^8 - 16*y^15 + 25*y^24 - 36*y^35 + 49*y^48 +...)
+ x^4*(1 - 5*y^4 + 14*y^10 - 30*y^18 + 55*y^28 - 91*y^40 + 140*y^54 +...)
+ x^5*(1 - 6*y^5 + 20*y^12 - 50*y^21 + 105*y^32 - 196*y^45 + 336*y^60 +...)
+ x^6*(1 - 7*y^6 + 27*y^14 - 77*y^24 + 182*y^36 - 378*y^50 + 714*y^66 +...)
+ x^7*(1 - 8*y^7 + 35*y^16 - 112*y^27 + 294*y^40 - 672*y^55 + 1386*y^72 +...)
+ x^8*(1 - 9*y^8 + 44*y^18 - 156*y^30 + 450*y^44 - 1122*y^60 + 2508*y^78 +...)
+...
Summing along columns gives the alternate g.f.:
A(x,y) = x/(1-x) + Sum_{n>=1} (-1)^n * x * y^(n^2) * (2 - x*y^n)/(1 - x*y^n)^(n+1).
Note that the coefficient of x in A(x,y) is Jacobi's theta_4 function of y.
Also, the coefficient of x^2 in A(x,y) equals Product_{n>=1} (1 - y^(2*n))^3.
RECTANGULAR ARRAY.
This array of coefficients T(n,k) of x^n * y^(k*(n+k-1)) in A(x,y) begins:
n=1: [1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, ...];
n=2: [1, -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, ...];
n=3: [1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, ...];
n=4: [1, -5, 14, -30, 55, -91, 140, -204, 285, -385, 506, ...];
n=5: [1, -6, 20, -50, 105, -196, 336, -540, 825, -1210, 1716, ...];
n=6: [1, -7, 27, -77, 182, -378, 714, -1254, 2079, -3289, 5005, ...];
n=7: [1, -8, 35, -112, 294, -672, 1386, -2640, 4719, -8008, 13013, ...];
n=8: [1, -9, 44, -156, 450, -1122, 2508, -5148, 9867, -17875, 30888, ...];
n=9: [1, -10, 54, -210, 660, -1782, 4290, -9438, 19305, -37180, 68068, ...]; ...
where row n has g.f.: (1 - z) / (1 + z)^n.
The array has the alternate g.f.: (1 - z) / (1 - x + z).
RELATED SERIES.
We may also write A(x,y) = P(x,y) + Q(x,y) where
P(x,y) = -1 + Sum_{n>=0} (-1)^n * y^(n*(n-1)) / (1 - x*y^n)^(n+1),
Q(x,y) = Sum_{n>=0} (-1)^n * y^(n*(n+1)) / (1 - x*y^(n+1))^n.
These series begin as follows:
P(x,y) = (-1 + y^2 - y^6 + y^12 - y^20 + y^30 - y^42 + y^56 - y^72 +...)
+ x*(1 - 2*y + 3*y^4 - 4*y^9 + 5*y^16 - 6*y^25 + 7*y^36 - 8*y^49 +...)
+ x^2*(1 - 3*y^2 + 6*y^6 - 10*y^12 + 15*y^20 - 21*y^30 + 28*y^42 +...)
+ x^3*(1 - 4*y^3 + 10*y^8 - 20*y^15 + 35*y^24 - 56*y^35 + 84*y^48 +...)
+ x^4*(1 - 5*y^4 + 15*y^10 - 35*y^18 + 70*y^28 - 126*y^40 + 210*y^54 +...)
+ x^5*(1 - 6*y^5 + 21*y^12 - 56*y^21 + 126*y^32 - 252*y^45 + 462*y^60 +...)
+ x^6*(1 - 7*y^6 + 28*y^14 - 84*y^24 + 210*y^36 - 462*y^50 + 924*y^66 +...)
+ x^7*(1 - 8*y^7 + 36*y^16 - 120*y^27 + 330*y^40 - 792*y^55 + 1716*y^72 +...)
+...
Q(x,y) = (1 - y^2 + y^6 - y^12 + y^20 - y^30 + y^42 - y^56 + y^72 +...)
+ x*(-y^4 + 2*y^9 - 3*y^16 + 4*y^25 - 5*y^36 + 6*y^49 - 7*y^64 +...)
+ x^2*(-y^6 + 3*y^12 - 6*y^20 + 10*y^30 - 15*y^42 + 21*y^56 +...)
+ x^3*(-y^8 + 4*y^15 - 10*y^24 + 20*y^35 - 35*y^48 + 56*y^63 +...)
+ x^4*(-y^10 + 5*y^18 - 15*y^28 + 35*y^40 - 70*y^54 + 126*y^70 +...)
+ x^5*(-y^12 + 6*y^21 - 21*y^32 + 56*y^45 - 126*y^60 + 252*y^77 +...)
+ x^6*(-y^14 + 7*y^24 - 28*y^36 + 84*y^50 - 210*y^66 + 462*y^84 +...)
+ x^7*(-y^16 + 8*y^27 - 36*y^40 + 120*y^55 - 330*y^72 + 792*y^91 +...)
+...

Crossrefs

Cf. A292929, A293385, A029635.

Programs

  • PARI
    { T(n,k) = my(z=x+x*O(x^k)); polcoeff( (1-z)/(1+z)^n, k) }
/* Print as a rectangular array: */
for(n=1,10,for(k=0,10,print1(T(n,k),", "));print(""))
/* Print as a triangle: */
for(n=0,14,for(k=0,n,print1(T(n-k+1,k),", "));print(""))
/* Print as a flattened array: */
for(n=0,14,for(k=0,n,print1(T(n-k+1,k),", "));)

Formula

G.f. A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1).
G.f. A(x,y) = x * Sum_{-oo..+oo} (x - y^n)^n.
G.f. A(x,y) = x/(1-x) + Sum_{n>=1} (-1)^n*x*y^(n^2)*(2 - x*y^n)/(1 - x*y^n)^(n+1).
G.f. A(x,y) = P(x,y) + Q(x,y) where
P(x,y) = Sum_{n>=0} (x - y^n)^(n+1),
P(x,y) = -1 + Sum_{n>=0} (-1)^n * y^(n*(n-1)) / (1 - x*y^n)^(n+1),
Q(x,y) = Sum_{n>=0} (-1)^n * y^(n*(n+1)) / (1 - x*y^(n+1))^n.
Showing 1-6 of 6 results.

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