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)

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).

A360771 Expansion of Sum_{k>=0} (x * (2 + x^k))^k.

Original entry on oeis.org

1, 2, 5, 8, 20, 32, 77, 128, 288, 518, 1104, 2048, 4313, 8192, 16832, 32848, 66568, 131072, 264688, 524288, 1053737, 2097824, 4205568, 8388608, 16803744, 33554442, 67162112, 134222336, 268550704, 536870912, 1073999165, 2147483648, 4295493376, 8589962752
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2023

Keywords

Crossrefs

Cf. A217668, A260148.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 2^(# - n/# + 1) * Binomial[#, n/# - 1] &]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Aug 02 2023 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(2+x^k))^k))
  • PARI
    a(n) = if(n==0, 1, sumdiv(n, d, 2^(d-n/d+1)*binomial(d, n/d-1)));

Formula

a(n) = Sum_{d|n} 2^(d-n/d+1) * binomial(d,n/d-1) for n > 0.
If p is an odd prime, a(p) = 2^p.

A360770 Expansion of Sum_{k>0} (x * (k + x^k))^k.

Original entry on oeis.org

1, 5, 27, 260, 3125, 46684, 823543, 16777472, 387420498, 10000003125, 285311670611, 8916100495009, 302875106592253, 11112006826381559, 437893890380860625, 18446744073726328848, 827240261886336764177, 39346408075296925015353
Offset: 1

Views

Author

Seiichi Manyama, Feb 20 2023

Keywords

Crossrefs

Cf. A260148, A338693, A360712, A360771.
Cf. A360774, A360775, A360776.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# - n/# + 1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (x*(k+x^k))^k))
  • PARI
    a(n) = sumdiv(n, d, d^(d-n/d+1)*binomial(d, n/d-1));

Formula

a(n) = Sum_{d|n} d^(d-n/d+1) * binomial(d,n/d-1).
If p is an odd prime, a(p) = p^p.
Showing 1-4 of 4 results.

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