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 1-4 of 4 results.

A260147 G.f. A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^n)^(2*n).

Original entry on oeis.org

1, 2, 1, 5, 1, 6, 8, 8, 1, 25, 12, 12, 29, 14, 36, 77, 1, 18, 151, 20, 71, 135, 166, 24, 121, 236, 287, 307, 30, 30, 1141, 32, 1, 727, 681, 1247, 314, 38, 970, 1652, 1821, 42, 2633, 44, 331, 6590, 1772, 48, 497, 3053, 7146, 6801, 1717, 54, 4051, 7427, 8009, 12389, 3655, 60, 17842, 62, 4496, 42841, 1, 15731, 6470, 68, 19449, 34754, 65781
Offset: 0

Views

Author

Paul D. Hanna, Jul 17 2015

Keywords

Comments

Compare to the curious identities:
(1) Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
(2) Sum_{n=-oo..+oo} (-x)^n * (1 + x^n)^n = 0.
Name changed for clarity by Paul D. Hanna, Dec 10 2024; prior name was "G.f.: (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function."

Examples

			G.f.: A(x) = 1 + 2*x^2 + x^4 + 5*x^6 + x^8 + 6*x^10 + 8*x^12 + 8*x^14 + x^16 + 25*x^18 + 12*x^20 + ...
where 2*A(x) = 1 + P(x) + N(x) with
P(x) = x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5 + ...
N(x) = 1/(1+x) + x^2/(1+x^2)^2 + x^6/(1+x^3)^3 + x^12/(1+x^4)^4 + x^20/(1+x^5)^5 + ...
Explicitly,
P(x) = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 + 4*x^9 + 6*x^10 + x^11 + 14*x^12 + x^13 + 8*x^14 + 11*x^15 + 13*x^16 + x^17 + 25*x^18 + x^19 + 22*x^20 + 22*x^21 + 12*x^22 + x^23 + 61*x^24 + 6*x^25 +...+ A217668(n)*x^n + ...
N(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 - x^11 + 2*x^12 - x^13 + 8*x^14 - 11*x^15 - 11*x^16 - x^17 + 25*x^18 - x^19 + 2*x^20 - 22*x^21 + 12*x^22 - x^23 - 3*x^24 - 6*x^25 +...+ A260148(n)*x^n + ...
From _Paul D. Hanna_, Dec 10 2024: (Start)
SPECIFIC VALUES.
A(z) = 0 at z = -0.404783857785183643579648014798209689698619095608142590080356...
  where 0 = Sum_{n=-oo..+oo} z^n * (1 + z^n)^(2*n).
A(t) = 8 at t = 0.66184860446935243758952792459096102121713616089603...
A(t) = 7 at t = 0.64280265347584821638335226655422639958638446962646...
A(t) = 6 at t = 0.61846293982236470622283664293769398297407552626520...
A(t) = 5 at t = 0.58591538561726828976301449562779896617926938759041...
A(t) = 4 at t = 0.53948212974289878102531393938569583066950526874204...
A(t) = 3 at t = 0.46633361832235508894561538442655261465230172977527...
A(t) = 2 at t = 0.33014122063168294490173944063355594394361494532642...
  where 2 = Sum_{n=-oo..+oo} t^n * (1 + t^n)^(2*n).
A(t) = -1 at t = -0.57221202613754835881500708971837082259712665852148...
A(t) = -2 at t = -0.66124771863833308133360587362156745037996654826889...
A(t) = -3 at t = -0.72841228559829175547612598129696947453305714538354...
A(t) = -4 at t = -0.90975449896027994776675798799643226140294233213401...
A(4/5) = 39.597156112579883800797829785472315940190856875500...
A(3/4) = 18.522637966827153559321082877260756270457362912092...
A(2/3) = 8.2917909754417331599016245586686519315443444070756...
A(3/5) = 5.3942577326786364433206097043093210828422082884565...
A(1/2) = 3.3971121875472777749836900920631175982646917998641...
  where A(1/2) = Sum_{n=-oo..+oo} (2^n + 1)^(2*n) / 2^(2*n^2+n).
A(2/5) = 2.4226617866265771206729430879848898772232404418272...
A(1/3) = 2.0164022766484546805373278337731916678136050742206...
  where A(1/3) = Sum_{n=-oo..+oo} (3^n + 1)^(2*n) / 3^(2*n^2+n).
A(1/4) = 1.6529591092151291503041860933179009814428123139546...
  where A(1/4) = Sum_{n=-oo..+oo} (4^n + 1)^(2*n) / 4^(2*n^2+n).
A(1/5) = 1.4841513733060571811336245213703004776194631749017...
  where A(1/5) = Sum_{n=-oo..+oo} (5^n + 1)^(2*n) / 5^(2*n^2+n).
(End)

Links

Crossrefs

Cf. A260116, A260148, A217668, A260180, A260361.
Cf. A261605, A363558, A363559, A363569, A363561.
Cf. A378573.

Programs

  • Maple
    with(numtheory):
seq(add(binomial(2*n/d, d-1) + (-1)^(n/d+1) * binomial(n/d, 2*d-1), d in divisors(n)), n = 1..70); # Peter Bala, Mar 02 2025
  • Mathematica
    terms = 100; max = 2 terms; 1/2 Sum[x^n*(1 + x^n)^n, {n, -max, max}] + O[x]^max // CoefficientList[#, x^2]& (* Jean-François Alcover, May 16 2017 *)
  • PARI
    {a(n) = my(A=1); A = sum(k=-2*n-2, 2*n+2, x^k*(1+x^k)^k/2 + O(x^(2*n+2)) ); polcoef(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); A = sum(k=-2*n-2, 2*n+2, x^(k^2-k) / (1 + x^k)^k /2  + O(x^(2*n+2)) ); polcoef(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); A = sum(k=-sqrtint(n)-1, n+1, x^k*((1+x^k)^(2*k) + (1-x^k)^(2*k))/2 + O(x^(n+1)) ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); A = sum(k=-n-1, n+1, x^k*(1+x^k)^(2*k) + O(x^(n+1)) ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=1); A = sum(k=-n-1, n+1, x^(2*k^2-k)/(1-x^k + O(x^(n+1)))^(2*k)  ); polcoef(A, n)}
for(n=0, 60, print1(a(n), ", "))

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^n)^n.
(2) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n.
(3) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n.
(4) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n.
(5) A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^n)^(2*n).
(6) A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^n)^(2*n).
(7) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - x^n)^(2*n).
(8) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + x^n)^(2*n).
a(2^n) = 1 for n > 0 (conjecture).
a(p) = p+1 for primes p > 3 (conjecture).
From Peter Bala, Jan 23 2021: (Start)
The following are conjectural:
A(x^2) = Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1) )^(2*n+1).
Equivalently: A(x^2) = Sum_{n = -oo..+oo} x^(4*n^2 + 2*n)/(1 + x^(2*n+1))^(2*n+1).
a(2*n+1) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(4*n+2)
More generally, for k = 1,2,3,..., a((2^k)*(2*n + 1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + x^(2*n+1))^(2^(k+1)*(2*n+1)).
a(2*n+1) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 + x^n)^(2*n) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 - x^n)^(2*n).
More generally, for k = 1,2,3,...,
a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 + x^n)^(2^(k+1)*n) = [x^(2*n+1)] Sum_{n = -oo..+oo} (-1)^(n+1)*x^n*(1 - x^n)^(2^(k+1)*n).
a(4*n+2) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 + x^n)^(2*n) = [x^(4*n+2)] Sum_{n = -oo..+oo} (-1)^n*x^n*(1 - x^n)^(2*n).
a(n) = [x^(2*n)] Sum_{n = -oo..+oo} (-1)^n*x^(2*n+1)*(1 + (-1)^n* x^(2*n+1) )^(2*n+1).
For k = 1,2,3,...,
a((2^k)*(2*n+1)) = [x^(2*n+1)] Sum_{n = -oo..+oo} x^(2*n+1)*(1 + (-1)^n* x^(2*n+1) )^(2^(k+1)*(2*n+1)).
(End)
From Peter Bala, Mar 02 2025: (Start)
a(n) = Sum_{d divides n} (binomial(2*n/d, d-1) + (-1)^(n/d+1) * binomial(n/d, 2*d-1)) for n >= 1.
Hence, a(p) = p + 1 for primes p > 3 and a(2^n) = 1 for n > 0 as conjectured above. (End)

A363558 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).

Original entry on oeis.org

1, 5, 16, 77, 256, 1104, 4121, 16832, 65536, 264688, 1048617, 4205568, 16779008, 67162112, 268436016, 1073999165, 4294967296, 17180983296, 68719549696, 274882887680, 1099511628896, 4398068904021, 17592186086656, 70368840646656, 281474978676736, 1125900326286464
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2023

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds as a formal power series for all y.

Examples

			G.f.: A(x) = 1 + 5*x + 16*x^2 + 77*x^3 + 256*x^4 + 1104*x^5 + 4121*x^6 + 16832*x^7 + 65536*x^8 + 264688*x^9 + 1048617*x^10 + ...

Links

Crossrefs

Cf. A260147, A363559, A363569, A363561.

Programs

  • PARI
    {a(n) = my(A); A = sum(m=-n-1,n+1, x^m * (2 + x^m +x*O(x^n))^(2*m) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1.a) A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).
(1.b) A(x) = Sum_{n=-oo..+oo} x^n * (2 - x^n)^(2*n).
(2.a) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - 2*x^n)^(2*n).
(2.b) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + 2*x^n)^(2*n).
(3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (2 + x^n)^n.
(3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (-2 + x^n)^n.
(4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + 2*x^n)^n.
(4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - 2*x^n)^n.
From Paul D. Hanna, Aug 06 2023: (Start)
The following generating functions are extensions of Peter Bala's formulas given in A260147.
(5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n+1) * (2 + x^(2*n+1))^(2*n+1).
(5.b) A(x^2) = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + 2*x^(2*n+1))^(2*n+1).
(End)
a(2^n) = 4^(2^n) for n > 0 (conjecture).
a(p) = p*2^(p-1) + 4^p for primes p > 3 (conjecture).

A363559 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).

Original entry on oeis.org

1, 10, 81, 757, 6561, 59454, 531496, 4788072, 43046721, 387480753, 3486784492, 31381709148, 282429556893, 2541872737062, 22876792457796, 205891204134565, 1853020188851841, 16677182431460826, 150094635300957591, 1350851725033981380, 12157665459056934471
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2023

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds as a formal power series for all y.

Examples

			G.f.: A(x) = 1 + 10*x + 81*x^2 + 757*x^3 + 6561*x^4 + 59454*x^5 + 531496*x^6 + 4788072*x^7 + 43046721*x^8 + 387480753*x^9 + ...

Links

Crossrefs

Cf. A260147, A363558, A363569, A363561.

Programs

  • PARI
    {a(n) = my(A); A = sum(m=-n-1,n+1, x^m * (3 + x^m +x*O(x^n))^(2*m) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1.a) A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).
(1.b) A(x) = Sum_{n=-oo..+oo} x^n * (3 - x^n)^(2*n).
(2.a) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - 3*x^n)^(2*n).
(2.b) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + 3*x^n)^(2*n).
(3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (3 + x^n)^n.
(3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (-3 + x^n)^n.
(4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + 3*x^n)^n.
(4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - 3*x^n)^n.
From Paul D. Hanna, Aug 06 2023: (Start)
The following generating functions are extensions of Peter Bala's formulas given in A260147.
(5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n+1) * (3 + x^(2*n+1))^(2*n+1).
(5.b) A(x^2) = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + 3*x^(2*n+1))^(2*n+1).
(End)
a(2^n) = 9^(2^n) for n > 0 (conjecture).
a(p) = p*3^(p-1) + 9^p for primes p > 3 (conjecture).

A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.

Original entry on oeis.org

2, 4, 2, 10, 2, 12, 16, 16, 2, 50, 24, 24, 58, 28, 72, 154, 2, 36, 302, 40, 142, 270, 332, 48, 242, 472, 574, 614, 60, 60, 2282, 64, 2, 1454, 1362, 2494, 628, 76, 1940, 3304, 3642, 84, 5266, 88, 662, 13180, 3544, 96, 994, 6106, 14292, 13602, 3434, 108, 8102, 14854, 16018, 24778, 7310, 120, 35684
Offset: 0

Views

Author

Paul D. Hanna, Jul 23 2015

Keywords

Comments

Compare to the curious identities:
(1) Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
(2) Sum_{n=-oo..+oo} (-x)^n * (1 + x^n)^n = 0.

Examples

			G.f.: A(x) = 2 + 4*x^2 + 2*x^4 + 10*x^6 + 2*x^8 + 12*x^10 + 16*x^12 + 16*x^14 + 2*x^16 + 50*x^18 + 24*x^20 +...
where A(x) = 1 + P(x) + N(x) with
P(x) = x*(1+x) + x^2*(1+x^2)^2 + x^3*(1+x^3)^3 + x^4*(1+x^4)^4 + x^5*(1+x^5)^5 +...
N(x) = 1/(1+x) + x^2/(1+x^2)^2 + x^6/(1+x^3)^3 + x^12/(1+x^4)^4 + x^20/(1+x^5)^5 +...
Explicitly,
P(x) = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 5*x^6 + x^7 + 5*x^8 + 4*x^9 + 6*x^10 + x^11 + 14*x^12 + x^13 + 8*x^14 + 11*x^15 + 13*x^16 + x^17 + 25*x^18 + x^19 + 22*x^20 + 22*x^21 + 12*x^22 + x^23 + 61*x^24 + 6*x^25 +...+ A217668(n)*x^n +...
N(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 - x^11 + 2*x^12 - x^13 + 8*x^14 - 11*x^15 - 11*x^16 - x^17 + 25*x^18 - x^19 + 2*x^20 - 22*x^21 + 12*x^22 - x^23 - 3*x^24 - 6*x^25 +...+ A260148(n)*x^n +...

Crossrefs

Cf. A260147, A217668, A260148, A363569, A363558, A363559, A363561.

Programs

  • Mathematica
    terms = 100; max = 2 terms; Sum[x^n*(1 + x^n)^n, {n, -max, max}] + O[x]^max // CoefficientList[#, x^2]& (* Jean-François Alcover, May 16 2017 *)
  • PARI
    {a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^k*(1+x^k)^k + O(x^(2*n+2)) ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1); A = sum(k=-2*n-2, 2*n+2, x^(k^2-k) / (1 + x^k)^k  + O(x^(2*n+2)) ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1); A = sum(k=1, sqrtint(2*n)+2, x^(k^2-k) *((1 + x^k)^k + (1 - x^k)^k) / (1 - x^(2*k)  + O(x^(2*n+2)) )^k ); polcoeff(A, 2*n)}
for(n=0, 60, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1); A = sum(k=-sqrtint(n)-1, n+1, x^k*((1+x^k)^(2*k) + (1-x^k)^(2*k)) + O(x^(n+1)) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

Formula

G.f.: Sum_{n=-oo..+oo} (-x)^n * (1 - x^n)^n.
G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n)^n.
G.f.: Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n)^n.
G.f.: Sum_{n>=1} x^(n^2-n) *((1 + x^n)^n + (1 - x^n)^n) / (1 - x^(2*n))^n.
G.f.: Sum_{n=-oo..+oo} x^n * ((1 + x^n)^(2*n) + (1 - x^n)^(2*n)) = Sum_{n>=0} a(n)*x^n.
a(n) = 2*A260147(n).
a(2^n) = 2 for n > 0 (conjecture).
a(p) = 2*p+2 for primes p > 3 (conjecture).
Showing 1-4 of 4 results.

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