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 1-3 of 3 results.

A293129 L.g.f.: Sum_{n=-oo..+oo} (x - x^(2*n-1))^(2*n-1) / (2*n-1).

Original entry on oeis.org

1, 4, 1, 15, 1, 12, 40, 16, 1, 77, 92, 24, 101, 28, 204, 373, 1, 36, 667, 40, 575, 689, 826, 48, 393, 1582, 1379, 1937, 590, 60, 6101, 64, 1, 5227, 3129, 9515, 1826, 76, 4390, 12404, 11341, 84, 18361, 88, 5875, 46320, 7844, 96, 1553, 33133, 38886, 50883, 25741, 108, 25507, 44993, 82265, 91449, 15835, 120, 150162, 124, 19376, 390653, 1, 104015, 29394, 136, 242217, 249506, 507789, 144, 210831, 148, 33079, 647187, 593029, 711482, 47101, 160
Offset: 1

Views

Author

Paul D. Hanna, Oct 11 2017

Keywords

Comments

Compare l.g.f. to: Sum_{n=-oo..+oo, n<>0} (x - x^n)^n / n = -log(1-x).
Here l.g.f. L(x) = Sum_{n>=1} a(n) * x^(2*n-1) / (2*n-1).
a(2^n + 1) = 1 for n >= 1 (conjecture).

Examples

			L.g.f.: L(x) = x + 4*x^3/3 + x^5/5 + 15*x^7/7 + x^9/9 + 12*x^11/11 + 40*x^13/13 + 16*x^15/15 + x^17/17 + 77*x^19/19 + 92*x^21/21 + 24*x^23/23 + 101*x^25/25 + 28*x^27/27 + 204*x^29/29 + 373*x^31/31 + x^33/33 + 36*x^35/35 + 667*x^37/37 + 40*x^39/39 + 575*x^41/41 + 689*x^43/43 + 826*x^45/45 + 48*x^47/47 + 393*x^49/49 + 1582*x^51/51 + 1379*x^53/53 + 1937*x^55/55 + 590*x^57/57 + 60*x^59/59 +...
such that L(x) = Sum_{n=-oo..+oo} (x - x^(2*n-1))^(2*n-1) / (2*n-1).
The coefficient of x^(2^n+1)/(2^n+1) in L(x) for n>=1 begins:
[4, 1, 1, 1, 1, 1, 1, 1, 1, ...],
and it appears that a(k) = 1 only at k = 1 and k = 2^n + 1 (n>=1).
We may write L(x) = P(x) + Q(x) where
P(x) = (x - x) + (x - x^3)^3/3 + (x - x^5)^5/5 + (x - x^7)^7/7 + (x - x^9)^9/9 + (x - x^11)^11/11 + (x - x^13)^13/13 + (x - x^15)^15/15 + (x - x^17)^17/17 + (x - x^19)^19/19 + (x - x^21)^21/21 +...+ (x - x^(2*n-1))^(2*n-1)/(2*n-1) +...
Q(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
Explicitly,
P(x) = x^3/3 - 4*x^5/5 + 8*x^7/7 - 11*x^9/9 + x^11/11 + 14*x^13/13 + x^15/15 - 50*x^17/17 + 58*x^19/19 + x^21/21 + x^23/23 - 54*x^25/25 + x^27/27 - 28*x^29/29 + 311*x^31/31 - 340*x^33/33 + x^35/35 + 75*x^37/37 + x^39/39 - 81*x^41/41 + 345*x^43/43 - 44*x^45/45 + x^47/47 - 1427*x^49/49 + 1531*x^51/51 - 52*x^53/53 + 496*x^55/55 - 1253*x^57/57 + x^59/59 + 1343*x^61/61 + x^63/63 - 2924*x^65/65 +...
Q(x) = x + 3*x^3/3 + 5*x^5/5 + 7*x^7/7 + 12*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + 51*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + 341*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + 2925*x^65/65 +...
The coefficient of x^(2^n+1)/(2^n+1) in P(x) for n>=1 begins:
[1, -4, -11, -50, -340, -2924, -169032, -33445208, -21619038032, 1 - A293599(n), ...].
The coefficient of x^(2^n+1)/(2^n+1) in Q(x) for n>=1 begins:
[3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, ..., A293599(n), ...].

Links

Crossrefs

Cf. A293597 (P(x)), A293598 (Q(x)), A293599, A291937.

Programs

  • PARI
    {a(n) = my(P,Q,Ox = O(x^(2*n+1)));
P = sum(m=1,n+1, (x - x^(2*m-1) +Ox)^(2*m-1) / (2*m-1) );
Q = sum(m=1,sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
(2*n-1)*polcoeff(P + Q, 2*n-1)}
for(n=1,80,print1(a(n),", "))

Formula

L.g.f.: Sum_{n=-oo..+oo} (x + x^(2*n-1))^(2*n-1) / (2*n-1) - note the plus sign.
L.g.f.: -log(1-x) - Sum_{n=-oo..+oo, n<>0} (x - x^(2*n))^(2*n) / (2*n).
L.g.f.: L(x) = P(x) + Q(x) where
P(x) = Sum_{n>=1} (x - x^(2*n-1))^(2*n-1) / (2*n-1),
Q(x) = Sum_{n>=1} x^((2*n-1)^2) / ( (2*n-1) * (1 - x^(2*n))^(2*n-1) ).

A293597 L.g.f.: Sum_{n>=1} (x - x^(2*n-1))^(2*n-1) / (2*n-1).

Original entry on oeis.org

0, 1, -4, 8, -11, 1, 14, 1, -50, 58, 1, 1, -54, 1, -28, 311, -340, 1, 75, 1, -81, 345, -44, 1, -1427, 1531, -52, 496, -1253, 1, 1343, 1, -2924, 738, -68, 9444, -10073, 1, -76, 1028, 3691, 1, -4691, 1, -6941, 21295, -92, 1, -55580, 33034, 28180, 1752, -11479, 1, -54063, 42847, 19437, 2186, -116, 1, -77934, 1, -124, 238507, -169032, 85151, -188859, 1, -25755, 3198, 432636, 1, -513328, 1, -148, 157041, -36005, 711327, -465347, 1
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2017

Keywords

Examples

			L.g.f.: A(x) = x^3/3 - 4*x^5/5 + 8*x^7/7 - 11*x^9/9 + x^11/11 + 14*x^13/13 + x^15/15 - 50*x^17/17 + 58*x^19/19 + x^21/21 + x^23/23 - 54*x^25/25 + x^27/27 - 28*x^29/29 + 311*x^31/31 - 340*x^33/33 + x^35/35 + 75*x^37/37 + x^39/39 - 81*x^41/41 + 345*x^43/43 - 44*x^45/45 + x^47/47 - 1427*x^49/49 + 1531*x^51/51 - 52*x^53/53 + 496*x^55/55 - 1253*x^57/57 + x^59/59 + 1343*x^61/61 + x^63/63 - 2924*x^65/65 +...
which may be written as
A(x) = (x - x) + (x - x^3)^3/3 + (x - x^5)^5/5 + (x - x^7)^7/7 + (x - x^9)^9/9 + (x - x^11)^11/11 + (x - x^13)^13/13 + (x - x^15)^15/15 + (x - x^17)^17/17 + (x - x^19)^19/19 + (x - x^21)^21/21 +...+ (x - x^(2*n-1))^(2*n-1)/(2*n-1) +...
The coefficient of x^(2^n+1)/(2^n+1) in A(x) for n>=1 begins:
[1, -4, -11, -50, -340, -2924, -169032, -33445208, -21619038032, 1 - A293599(n), ...].

Links

Crossrefs

Cf. A293129, A293598, A293599.

Programs

  • PARI
    {a(n) = my(A, Ox = O(x^(2*n+1)));
A = sum(m=1, n+1, (x - x^(2*m-1) +Ox)^(2*m-1) / (2*m-1) );
(2*n-1)*polcoeff(A, 2*n-1)}
for(n=1, 80, print1(a(n), ", "))

A293598 L.g.f.: Sum_{n>=1} x^((2*n-1)^2) / ( (2*n-1) * (1 - x^(2*n))^(2*n-1) ).

Original entry on oeis.org

1, 3, 5, 7, 12, 11, 26, 15, 51, 19, 91, 23, 155, 27, 232, 62, 341, 35, 592, 39, 656, 344, 870, 47, 1820, 51, 1431, 1441, 1843, 59, 4758, 63, 2925, 4489, 3197, 71, 11899, 75, 4466, 11376, 7650, 83, 23052, 87, 12816, 25025, 7936, 95, 57133, 99, 10706, 49131, 37220, 107, 79570, 2146, 62828, 89263, 15951, 119, 228096, 123, 19500, 152146, 169033, 18864, 218253, 135, 267972, 246308, 75153, 143, 724159, 147, 33227, 490146, 629034, 155, 512448, 159
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2017

Keywords

Examples

			L.g.f.: A(x) = x + 3*x^3/3 + 5*x^5/5 + 7*x^7/7 + 12*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + 51*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + 341*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + 2925*x^65/65 +...
which may be written as
A(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
The coefficient of x^(2^n+1)/(2^n+1) in A(x) for n>=1 begins:
[3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, ..., A293599(n), ...].

Links

Crossrefs

Cf. A293129, A293597, A293599.

Programs

  • Mathematica
    nmax = 80; Table[(CoefficientList[Series[Sum[x^((2*k - 1)^2)/((2*k - 1)*(1 - x^(2*k))^(2*k - 1)), {k, 1, 2*nmax + 1}], {x, 0, 2*nmax + 1}], x] * Range[0, 2*nmax + 1])[[2*n]], {n, 1, nmax}] (* Vaclav Kotesovec, Oct 15 2017 *)
  • PARI
    {a(n) = my(A, Ox = O(x^(2*n+1)));
A = sum(m=1, sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
(2*n-1)*polcoeff(A, 2*n-1)}
for(n=1, 80, print1(a(n), ", "))
Showing 1-3 of 3 results.

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