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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.

Showing 1-3 of 3 results.

A293599 The coefficient of x^(2^n+1)/(2^n+1) in the l.g.f. of A293598 for n>=1.

Original entry on oeis.org

3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, 3270933679995185, 344648907850020294305, 20381496562418327375031168210529, 303229033555187108276527297692992345985345, 533360801574481336406792124161160375221861972273961952144925889, 331572178130571824652402094592695034861147899073590997231695381294750188182312600193
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2017

Keywords

Comments

The l.g.f. of A293598 is Sum_{n>=1} 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 the l.g.f. of A293597 equals 1 - a(n) for n>=2.
What is the rate of growth of this sequence?

Examples

			L.g.f. of A293598: 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,
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 +...
This sequence equals the coefficient of x^(2^n+1)/(2^n+1) in Q(x) for n>=1.

Links

Crossrefs

Cf. A293129, A293597, A293598.

Programs

  • Mathematica
    nmax = 10; 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 + 2]], {n, 1, nmax}] (* Vaclav Kotesovec, Oct 15 2017 *)
  • PARI
    {A293598(n) = my(Q, Ox = O(x^(2*n+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(Q, 2*n-1)}
for(n=0, 15, print1(A293598(2^n+1), ", "))

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), ", "))
Showing 1-3 of 3 results.

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