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)

A290003 G.f.: A(x) = Sum_{n=-oo..+oo} (x - x^n)^n.

Original entry on oeis.org

1, -1, 1, -2, 3, -3, 1, 1, 1, -7, 10, -6, 1, 0, 1, -8, 23, -25, 1, 17, 1, -32, 36, -12, 1, -21, 26, -14, 55, -92, 1, 93, 1, -129, 78, -18, 108, -121, 1, -20, 105, -49, 1, 19, 1, -298, 430, -24, 1, -423, 50, 424, 171, -469, 1, -217, 661, -450, 210, -30, 1, -203, 1, -32, 591, -897, 1288, -881, 1, -987, 300, 2407, 1, -2804, 1, -38, 2626, -1350, 1387, -2380, 1, 837, 487, -42, 1, -2855, 3741, -44, 465, -3301, 1, -326, 4291, -2324, 528, -48, 5815, -12713, 1, 6957, 1422, 4074, 1, -10371, 1, -8451, 20322, -54, 1, -15589, 1
Offset: 0

Views

Author

Paul D. Hanna, Sep 03 2017

Keywords

Comments

Compare g.f. to: Sum_{n=-oo..+oo} (x - x^(n+1))^n = 0.

Examples

			G.f.: A(x) = 1 - x + x^2 - 2*x^3 + 3*x^4 - 3*x^5 + x^6 + x^7 + x^8 - 7*x^9 + 10*x^10 - 6*x^11 + x^12 + x^14 - 8*x^15 + 23*x^16 - 25*x^17 + x^18 + 17*x^19 + x^20 - 32*x^21 + 36*x^22 - 12*x^23 + x^24 - 21*x^25 + 26*x^26 - 14*x^27 + 55*x^28 - 92*x^29 + x^30 +...
where A(x) = 1 + P(x) + N(x) with
P(x) = (x-x) + (x-x^2)^2 + (x-x^3)^3 + (x-x^4)^4 + (x-x^5)^5 + (x-x^6)^6 + (x-x^7)^7 +...+ (x-x^n)^n +...
N(x) = -x/(1 - x^2) + x^4/(1-x^3)^2 - x^9/(1-x^4)^3 + x^16/(1-x^5)^4 - x^25/(1-x^6)^5 +...+ (-x)^(n^2)/(1-x^(n+1))^n +...
Explicitly,
P(x) = x^2 - x^3 + 2*x^4 - 2*x^5 + x^6 + x^8 - 5*x^9 + 7*x^10 - 5*x^11 + x^12 + x^14 - 7*x^15 + 17*x^16 - 18*x^17 + x^18 + 12*x^19 + x^20 - 25*x^21 + 29*x^22 - 11*x^23 + x^24 - 12*x^25 + 16*x^26 - 13*x^27 + 46*x^28 - 70*x^29 + x^30 +...
N(x) = -x - x^3 + x^4 - x^5 + x^7 - 2*x^9 + 3*x^10 - x^11 - x^15 + 6*x^16 - 7*x^17 + 5*x^19 - 7*x^21 + 7*x^22 - x^23 - 9*x^25 + 10*x^26 - x^27 + 9*x^28 - 22*x^29 +...
From _Paul D. Hanna_, Jan 13 2025: (Start)
SPECIAL VALUES.
A local maximum of A(x) is at x = z, A'(z) = 0,
  where z = 0.6783626505745664596168958924200373689742586374321477329...
  and A(z) = 0.332320805615430858829730480236535256165083297416146964...
A(5/6) = 0.30801526795391347776371668529063511729774504098314...
A(4/5) = 0.31797024517441016604092565708098992009134940089362...
A(3/4) = 0.32759707660987407896902126812991555844980484348844...
A(2/3) = 0.33220302782561874934924055409715505666564907222676...
A(3/5) = 0.32724657183605678719721082848286562112862495149949...
A(1/2) = 0.30725396830704316799197832656390411971168116373389...
A(2/5) = 0.27337943400586708871078028747061201307317280175586...
A(1/3) = 0.24338606674563424484910361835257533242309621632065...
A(1/4) = 0.19758524006807690544490179709803177425355852401229...
A(1/5) = 0.16558333624735433324843855679493132539350188690309...
A(1/6) = 0.14230098666491512550971306545368484826875874989347...
(End)

Links

Crossrefs

Cf. A260116, A260147, A378582, A379195.

Programs

  • Maple
    with(numtheory):
seq(add((-1)^(d-1) * (binomial((n-1)/d, d-2) + binomial((n+d-1)/d, d-1)), d in divisors(n-1)), n = 2..110); # Peter Bala, Mar 02 2025
  • PARI
    {a(n) = local(A=1); A = sum(k=-n, n, (x - x^k)^k +x*O(x^n)); polcoeff(A, n)}
for(n=0, 100, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1); A = 1 + sum(k=1, n, x^k*(1 - x^(k-1))^k + (-x)^(k^2)/(1 - x^(k+1))^k +x*O(x^n)); polcoeff(A, n)}
for(n=0, 500, print1(a(n), ", "))

Formula

G.f.: 1 + Sum_{n>=1} x^n*(1 - x^(n-1))^n + (-x)^(n^2)/(1 - x^(n+1))^n.
a(p+1) = 1 for primes p > 3 (conjecture).
From Peter Bala, Mar 02 2025: (Start)
The above conjecture follows from the following formula: for n >= 2,
a(n) = Sum_{d divides n-1} (-1)^(d-1) * ( binomial((n-1)/d, d-2) + binomial((n+d-1)/d, d-1) ).
For prime p >= 3, a(p^2 + 1) = p^2 + 1. (End)

A260180 G.f.: Sum_{n>=0} x^n * (1 - x^n)^n.

Original entry on oeis.org

1, 1, 0, 1, -1, 1, -1, 1, -3, 4, -4, 1, 0, 1, -6, 11, -11, 1, 7, 1, -18, 22, -10, 1, -3, 6, -12, 37, -48, 1, 45, 1, -71, 56, -16, 36, -41, 1, -18, 79, -69, 1, 51, 1, -186, 232, -22, 1, -179, 8, 186, 137, -311, 1, 1, 331, -364, 172, -28, 1, -51, 1, -30, 295, -599, 716, -263, 1, -713, 254, 1177, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 17 2015

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + x^3 - x^4 + x^5 - x^6 + x^7 - 3*x^8 + 4*x^9 - 4*x^10 +...
where
A(x) = 1 + 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 +...
Also,
A(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 +...

Links

Crossrefs

Cf. A260116, A217668, A260147.

Programs

  • Maple
    with(numtheory):
seq(add((-1)^(d-1)*binomial(n/d, d-1), d in divisors(n)), n = 1..70); # Peter Bala, Mar 02 2025
  • Mathematica
    terms = 100; 1 + Sum[x^n*(1 - x^n)^n, {n, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)
  • PARI
    {a(n) = local(A=1); A = sum(k=0, n+1, x^k*(1-x^k)^k + O(x^(n+2))); polcoeff(A, n)}
for(n=0, 80, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1); A = sum(k=1, n+1, -1/x^k / (1 - 1/x^k + O(x^(n+2)) )^k + O(x^(n+2))); polcoeff(A, n)}
for(n=0, 80, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1); A = sum(k=1, sqrtint(n)+1, (-1)^(k-1) * x^(k^2-k)/(1-x^k)^k + O(x^(n+2))); polcoeff(A, n)}
for(n=0, 80, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=1} (-1)^(n-1) * x^(n^2-n) / (1 - x^n)^n.
G.f.: Sum_{n>=1} - x^(-n) / (1 - x^(-n))^n.
From Peter Bala, Mar 02 2025: (Start)
For n >= 1, a(n) = Sum_{d divides n} (-1)^(d-1) * binomial(n/d, d-1).
For prime p > 3, a(p) = 1, a(2*p) = 1 - p and a(p^2) = p + 1. (End)

A354646 G.f. A(x) satisfies: 1 = Sum_{n=-oo..+oo} (x + x^n)^n * (-2*A(x))^(n*(n-1)/2).

Original entry on oeis.org

1, -1, -4, 44, 316, -22695, -769536, 156937802, 30299780744, -18827264809946, -17187430890378027, 37887447329364481223, 148620374587239353630657, -1249806569497062808351943525, -20168103472406206381500342351035, 666759209181977763318463790517458280
Offset: 0

Views

Author

Paul D. Hanna, Jun 07 2022

Keywords

Examples

			G.f.: A(x) = 1 - x - 4*x^2 + 44*x^3 + 316*x^4 - 22695*x^5 - 769536*x^6 + 156937802*x^7 + 30299780744*x^8 - 18827264809946*x^9 - 17187430890378027*x^10 ++-- ...
such that
B(x) = Sum_{n>=1} x^n * (1 + x^(n-1))^n * (-2*A(x))^(n*(n-1)/2)
and
B(x) = -Sum_{n>=1} x^(n^2) / (1 + x^(n+1))^n * (-2*A(x))^(n*(n+1)/2),
where
B(x) = 2*x - 2*x^2 - 10*x^3 + 98*x^4 + 618*x^5 - 45552*x^6 - 1538490*x^7 + 313926892*x^8 + 60600533658*x^9 - 37654860921240*x^10 - 34374918573912040*x^11 ++-- ...

Links

Crossrefs

Cf. A260116.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff(sum(m=-#A,#A, (x + x^m)^m * (-2*Ser(A))^(m*(m-1)/2) ),#A)/2);H=A;A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (1 + x^(n-1))^n * (-2*A(x))^(n*(n-1)/2).
(2) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 + x^(n+1))^n * (-2*A(x))^(n*(n+1)/2).
Showing 1-4 of 4 results.

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