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 1-10 of 15 results. Next

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).

A363569 Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1.

Original entry on oeis.org

1, 0, 1, -3, 1, 4, -4, -8, 1, 23, -8, -12, -27, 12, 36, 15, 1, 16, -149, -20, 71, 91, 166, -24, -119, -186, -285, 251, -26, 28, 761, -32, 1, -199, -679, -827, -310, 36, 970, 1572, 1821, 40, -2631, -44, 331, -5628, 1772, -48, -495, 3051, 2546, 6697, -1715, 52, 2791, -7425, -8007, -10869, -3653, -60
Offset: 0

Views

Author

Paul D. Hanna, Jul 31 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 + x^2 - 3*x^3 + x^4 + 4*x^5 - 4*x^6 - 8*x^7 + x^8 + 23*x^9 - 8*x^10 - 12*x^11 - 27*x^12 + 12*x^13 + 36*x^14 + 15*x^15 + ...

Links

Crossrefs

Cf. A260147, A363558, A363559.

Programs

  • PARI
    {a(n) = my(A); A = sum(m=-n-1,n+1, x^m * (I + x^m +x*O(x^n))^(2*m) ); polcoeff(A,n)}
for(n=0,60,print1(a(n),", "))

Formula

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

A260148 Expansion of Sum_{n>=0} x^(n^2-n) / (1 + x^n)^n.

Original entry on oeis.org

1, -1, 2, -1, -1, -1, 5, -1, -3, -4, 6, -1, 2, -1, 8, -11, -11, -1, 25, -1, 2, -22, 12, -1, -3, -6, 14, -37, 22, -1, 77, -1, -71, -56, 18, -36, 127, -1, 20, -79, -69, -1, 135, -1, 144, -232, 24, -1, -179, -8, 236, -137, 261, -1, 307, -331, -362, -172, 30, -1, 859, -1, 32, -295, -599, -716, 727, -1, 647, -254, 1247
Offset: 0

Views

Author

Paul D. Hanna, Jul 17 2015

Keywords

Examples

			A(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 +...
where
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 + ...
Also,
A(x) = 1 + x*(x-1) + x^2*(x^2-1)^2 + x^3*(x^3-1)^3 + x^4*(x^4-1)^4 + x^5*(x^5-1)^5 + ...

Links

Crossrefs

Cf. A217668, A260147.

Programs

  • PARI
    a(n) = local(A=1); A = sum(k=1, sqrtint(n)+1, x^(k^2-k) / (1 + x^k +x*O(x^n) )^k ); polcoeff(A, n);
for(n=0, 70, print1(a(n), ", "))
  • PARI
    a(n) = local(A=0); A = sum(k=1,n+1, x^(-k)/(1 + x^(-k) +x*O(x^n) )^k ); polcoeff(A, n);
for(n=0, 70, print1(a(n), ", "))
  • PARI
    a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-n/d+1)*binomial(d, n/d-1))); \\ Seiichi Manyama, Feb 20 2023

Formula

G.f.: Sum_{n>=1} x^(-n) / (1 + x^(-n))^n.
G.f.: Sum_{n>=0} x^n * (x^n - 1)^n. - Paul D. Hanna, May 30 2018
a(p) = -1 for odd primes p.
a(n) = Sum_{d|n} (-1)^(d-n/d+1) * binomial(d,n/d-1) for n > 0. - Seiichi Manyama, Feb 20 2023

A261605 G.f.: Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n)^n.

Original entry on oeis.org

1, 2, -1, 4, -1, 6, -6, 8, 2, 12, -15, 12, 8, 14, -28, 32, 22, 18, -55, 20, 34, 72, -66, 24, 44, 28, -91, 140, 62, 30, -205, 32, 209, 244, -153, 72, -98, 38, -190, 392, 443, 42, -518, 44, -1, 788, -276, 48, 506, 52, -451, 852, -196, 54, -1086, 728, 1636, 1180, -435, 60, -1691
Offset: 0

Views

Author

Paul D. Hanna, Aug 25 2015

Keywords

Examples

			G.f.: A(x) = 1 + 2*x - x^2 + 4*x^3 - x^4 + 6*x^5 - 6*x^6 + 8*x^7 + 2*x^8 + 12*x^9 - 15*x^10 + 12*x^11 + 8*x^12 + 14*x^13 - 28*x^14 + 32*x^15 + 22*x^16 +...
where A(x) = 1 + N(x) + P(x) such that
N(x) = (x-1) + (x^2-1)^2 + (x^3-1)^3 + (x^4-1)^4 + (x^5-1)^5 + (x^6-1)^6 +...
P(x) = x/(1-x) + x^4/(1-x^2)^2 + x^9/(1-x^3)^3 + x^16/(1-x^4)^4 + x^25/(1-x^5)^5 +...
explicitly,
N(x) = x - 2*x^2 + 3*x^3 - 3*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 2*x^8 + 10*x^9 - 20*x^10 + 11*x^11 - x^12 + 13*x^13 - 35*x^14 + 25*x^15 + 13*x^16 +...
P(x) = x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 5*x^10 + x^11 + 9*x^12 + x^13 + 7*x^14 + 7*x^15 + 9*x^16 +...+ A143862(n)*x^n +...

Links

Crossrefs

Cf. A143862, A260147, A261608.

Programs

  • Maple
    with(numtheory):
seq(add( (-1)^(n/d+d)*binomial(d, n/d) + binomial(n/d-1, d-1), d in divisors(n)), n = 1..60); # Peter Bala, Mar 03 2025
  • PARI
    {a(n) = polcoeff(sum(m=-n-2,n+2,x^(m^2)/(1-x^m +x*O(x^n))^m), n)}
for(n=0,60,print1(a(n),", "))
  • PARI
    {a(n) = polcoeff(sum(m=-n-2,n+2,(x^m-1 +x*O(x^n))^m), n)}
for(n=0,60,print1(a(n),", "))
  • PARI
    {a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^(m^2)/(1+x^m +x*O(x^n))^(m+1)), n)}
for(n=0,60,print1(a(n),", "))
  • PARI
    {a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^m*(1+x^m +x*O(x^n))^(m-1)), n)}
for(n=0,60,print1(a(n),", "))

Formula

G.f.: Sum_{n=-oo..+oo} (x^n - 1)^n.
G.f.: 1/2 + Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n)^(n+1).
G.f.: 1/2 + Sum_{n=-oo..+oo} x^n * (1 + x^n)^(n-1).
From Peter Bala, Mar 03 2025: (Start)
For n >= 1, a(n) = Sum_{d divides n} (-1)^(n/d+d)*binomial(d, n/d) + binomial(n/d-1, d-1).
For odd prime p, a(p) = p + 1, a(2*p) = - p*(p + 1)/2, a(p^2) = p^2 + 3. (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)

A363561 G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).

Original entry on oeis.org

1, 3, 4, 15, 16, 52, 77, 184, 256, 716, 1045, 2400, 4320, 9024, 16524, 35439, 65536, 135424, 264928, 534016, 1048856, 2124523, 4196944, 8435712, 16792576, 33658512, 67118016, 134478584, 268435513, 537346048, 1073876144, 2148499456, 4294967296, 8592337520, 17179956224, 34364358760
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 + 3*x + 4*x^2 + 15*x^3 + 16*x^4 + 52*x^5 + 77*x^6 + 184*x^7 + 256*x^8 + 716*x^9 + 1045*x^10 + 2400*x^11 + 4320*x^12 + ...

Links

Crossrefs

Cf. A260147, A363558, A363559, A363569.

Programs

  • PARI
    {a(n) = my(A); A = sum(m=-n-1, n+1, x^m * (sqrt(2) + x^m +x*O(x^n))^(2*m) ); polcoeff(A, n)}
for(n=0, 30, print1(round(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 * (sqrt(2) + x^n)^(2*n).
(1.b) A(x) = Sum_{n=-oo..+oo} x^n * (sqrt(2) - x^n)^(2*n).
(2.a) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - sqrt(2)*x^n)^(2*n).
(2.b) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + sqrt(2)*x^n)^(2*n).
(3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^n.
(3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (-sqrt(2) + x^n)^n.
(4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + sqrt(2)*x^n)^n.
(4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - sqrt(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) * (sqrt(2) + x^(2*n+1))^(2*n+1).
(5.b) A(x^2) = Sum_{n=-oo..+oo} x^(2*n*(2*n+1)) / (1 + sqrt(2)*x^(2*n+1))^(2*n+1).
(End)
a(2^n) = 2^(2^n) for n > 0 (conjecture).
a(p) = p*2^((p-1)/2) + 2^p for primes p > 3 (conjecture).

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)

A262007 G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n / (1 - x)^n.

Original entry on oeis.org

1, 2, 1, 8, 7, 27, 45, 102, 194, 439, 844, 1775, 3608, 7342, 14891, 30283, 61113, 123625, 249355, 502430, 1011305, 2034028, 4086860, 8206874, 16469851, 33035697, 66234208, 132746099, 265961186, 532718115, 1066778721, 2135822309, 4275459594, 8557335615, 17125445126, 34268966022, 68568212859, 137187104632
Offset: 1

Views

Author

Paul D. Hanna, Sep 21 2015

Keywords

Comments

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
Compare also to the g.f. of A077229, where A077229(n) equals the number of compositions of n where the largest part is <= the number of parts.

Examples

			G.f.: A(x) = x + 2*x^2 + x^3 + 8*x^4 + 7*x^5 + 27*x^6 + 45*x^7 + 102*x^8 + 194*x^9 + 439*x^10 + 844*x^11 + 1775*x^12 +...
such that A(x) = N(x) + P(x) where
N(x) = Sum_{n>=1} (-1)^n * x^(n^2-n) * (1 - x)^n / (1 - x^n)^n
P(x) = Sum_{n>=0} x^n * (1 - x^n)^n / (1 - x)^n.
Explicitly,
N(x) = -1 + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 4*x^6 - 3*x^7 + 4*x^8 - 10*x^9 + 18*x^10 - 19*x^11 + 9*x^12 + 2*x^13 + x^14 - 22*x^15 + 50*x^16 +...
P(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 23*x^6 + 48*x^7 + 98*x^8 + 204*x^9 + 421*x^10 + 863*x^11 + 1766*x^12 + 3606*x^13 + 7341*x^14 + 14913*x^15 + 30233*x^16 +...+ A077229(n)*x^n +...

Links

Crossrefs

Cf. A077229, A260147.

Programs

  • PARI
    {a(n) = local(A=1);
A = sum(k=-n-1, n+1, x^k * (1-x^k)^k / (1-x +x*O(x^n))^k); polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=1);
A = sum(k=-n-1, n+1, (-1)^k * x^(k^2-k) * (1 - x)^k / (1 - x^k +x*O(x^n))^k); polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))

Formula

G.f.: Sum_{n=-oo..+oo} (-1)^n * x^(n^2-n) * (1 - x)^n / (1 - x^n)^n.
Limit a(n)^(1/n) = 2.
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 03 2017

A378573 G.f. A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^(3*n+1))^(2*n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 1, 1, 1, 8, 7, 1, 2, 10, 1, 1, 1, 27, 1, 1, 1, 14, 11, 1, 30, 16, 1, 1, 1, 18, 1, 46, 36, 20, 1, 1, 1, 37, 67, 2, 1, 24, 85, 1, 1, 117, 1, 1, 1, 28, 71, 1, 286, 30, 1, 22, 1, 33, 1, 154, 1, 34, 287, 211, 1, 36, 191, 1, 1, 38, 1, 127, 456, 271, 1, 1, 524, 42, 2, 1, 277, 44, 681, 1, 1, 46, 1, 788, 1, 1049
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2024

Keywords

Comments

Related identities:
(C.1) Sum_{n=-oo..+oo} x^n * (1 - x^(3*n+2))^n = 0.
(C.2) Sum_{n=-oo..+oo} (-1)^n * x^(3*n*(n+1)) / (1 + x^(3*n+1))^(n+1) = 0.

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + x^4 + x^5 + x^6 + 4*x^7 + x^8 + 2*x^9 + x^10 + 6*x^11 + x^12 + x^13 + x^14 + 8*x^15 + 7*x^16 + x^17 + 2*x^18 + 10*x^19 + x^20 + ...
SPECIFIC VALUES.
A(z) = 0 at z = -0.6726473467784327964946394402158022892169850805633511277...
  where 0 = Sum_{n=-oo..+oo} z^n * (1 + z^(3*n+1))^(2*n).
A(t) = 8 at t = 0.80674137409155594738508715662274076269252097031895...
A(t) = 7 at t = 0.79012273526862596166723863415319642411267133718829...
A(t) = 6 at t = 0.76819406763538484112048712466638978472377443909212...
A(t) = 5 at t = 0.73777899222025289918616588954081072720456797874020...
A(t) = 4 at t = 0.69251246918071024586564098631094327512630629569865...
A(t) = 3 at t = 0.61751935356221793541340938213050415525442896744761...
A(t) = 2 at t = 0.46815043155347172312205584241722605840913217439574...
  where 2 = Sum_{n=-oo..+oo} t^n * (1 + t^(3*n+1))^(2*n).
A(t) = -1 at t = -0.77517567890012104592411512614387150563591857093990...
A(t) = -2 at t = -0.81854774961928757410155043510790044331007733543405...
a(t) = -3 at t = -0.84450708995424907597930320956281576983716334202613...
A(4/5) = 7.5631342681464228254307790972990507013398446615499...
A(3/4) = 5.3598446737980629233504982857608095266005392614233...
A(2/3) = 3.5884575039557777965471951540301884270744196834272...
A(3/5) = 2.8340662386949213795469239985484973660637713412934...
A(1/2) = 2.1531614564039262021396751059639076614616014159933...
  where A(1/2) = Sum_{n=-oo..+oo} (2^(3*n+1) + 1)^(2*n) / 2^(3*n*(2*n+1)).
A(2/5) = 1.7360641537921941524470509621075633132346504795101...
A(1/3) = 1.5384884473879487136671091866260679901472537410267...
  where A(1/3) = Sum_{n=-oo..+oo} (3^(3*n+1) + 1)^(2*n) / 3^(3*n*(2*n+1)).
A(1/4) = 1.3491464535561504459384378489663997806149645910211...
  where A(1/4) = Sum_{n=-oo..+oo} (4^(3*n+1) + 1)^(2*n) / 4^(3*n*(2*n+1)).
A(1/5) = 1.2580390146694337862857122246948093151986010024710...
A(-1/3) = 0.71151183654243437744829702888914217294561469541322...
A(-1/2) = 0.51369607391129963764388587069816692146820022971981...
A(-2/3) = 0.03171944560249247956083537535107529561830519903038...

Links

Crossrefs

Cf. A260147.

Programs

  • PARI
    {a(n) = my(A = sum(m=-n,n, x^m * (1 + x^(3*m+1) +x*O(x^n))^(2*m) )); polcoef(A,n)}
for(n=0,100, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = Sum_{n=-oo..+oo} x^n * (1 + x^(3*n+1))^(2*n).
(1.b) A(x) = Sum_{n=-oo..+oo} x^n * (1 - x^(3*n+1))^(2*n).
(2.a) A(x) = Sum_{n=-oo..+oo} x^(3*n*(2*n-1)) / (1 + x^(3*n-1))^(2*n).
(2.b) A(x) = Sum_{n=-oo..+oo} x^(3*n*(2*n-1)) / (1 - x^(3*n-1))^(2*n).
(3.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^n * (1 + x^(3*n+2))^n.
(3.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-1)^n * x^n * (1 - x^(3*n+2))^n.
(4.a) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(3*n*(n+1)) / (1 + x^(3*n+1))^(n+1).
(4.b) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(3*n*(n+1)) / (1 - x^(3*n+1))^(n+1).
(5.a) A(x^2) = Sum_{n=-oo..+oo} x^(2*n-1) * (1 + x^(6*n-1))^(2*n-1).
(5.b) A(x^2) = -Sum_{n=-oo..+oo} x^(2*n-1) * (1 - x^(6*n-1))^(2*n-1).
(6.a) A(x^2) = Sum_{n=-oo..+oo} x^(6*n*(2*n+1)) / (1 + x^(6*n+1))^(2*n+1).
(6.b) A(x^2) = Sum_{n=-oo..+oo} x^(6*n*(2*n+1)) / (1 - x^(6*n+1))^(2*n+1).
Showing 1-10 of 15 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.