A217668 -id:A217668 - cp's OEIS frontend

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

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

Paul D. Hanna, Jul 17 2015

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."

			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)

Paul D. Hanna, Table of n, a(n) for n = 0..8200

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

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

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 *)

{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), ", "))

{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), ", "))

{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), ", "))

{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), ", "))

{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), ", "))

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)

A217670 G.f.: Sum_{n>=0} x^n/(1 + x^n)^n.

Original entry on oeis.org

1, 1, 0, 2, -2, 2, 0, 2, -8, 8, 0, 2, -12, 2, 0, 32, -36, 2, 0, 2, -20, 58, 0, 2, -136, 72, 0, 92, -28, 2, 0, 2, -272, 134, 0, 422, -288, 2, 0, 184, -480, 2, 0, 2, -44, 1232, 0, 2, -2360, 926, 0, 308, -52, 2, 0, 2004, -1176, 382, 0, 2, -4064, 2, 0, 6470, -5128, 3642

Offset: 0

Paul D. Hanna, Oct 10 2012

sign

			G.f.: A(x) = 1 + x + 2*x^3 - 2*x^4 + 2*x^5 + 2*x^7 - 8*x^8 + 8*x^9 +...
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 +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)

Cf. A048272, A157019, A217668.

terms = 100; Sum[x^n/(1 + x^n)^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)

{a(n)=polcoeff(sum(m=0, n, x^m/(1+x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))

a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-2, d-1))); \\ Seiichi Manyama, Apr 23 2021

a(4*n+2) = 0 for n>=0.
From Seiichi Manyama, Apr 23 2021: (Start)
a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-2, d-1) for n > 0.
If p is prime, a(p) = 1 + (-1)^(p-1). (End)

A217669 G.f.: Sum_{n>=0} (x + x^n)^n.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 1, 8, 1, 7, 7, 7, 1, 22, 1, 9, 17, 20, 1, 32, 1, 37, 29, 13, 1, 86, 16, 15, 46, 72, 1, 113, 1, 102, 67, 19, 72, 239, 1, 21, 92, 313, 1, 191, 1, 244, 331, 25, 1, 575, 29, 357, 154, 392, 1, 452, 496, 577, 191, 31, 1, 1979, 1, 33, 443, 750, 1002

Offset: 0

Paul D. Hanna, Oct 10 2012

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),
for any fixed integer k; here, k = 1 and y = 1, F = x, G = x.

			G.f.: A(x) = 1 + 2*x + x^2 + 3*x^3 + 2*x^4 + 4*x^5 + x^6 + 8*x^7 + x^8 +...
where
A(x) = 1 + (x + x) + (x + x^2)^2 + (x + x^3)^3 + (x + x^4)^4 + (x + x^5)^5 +...
Also
A(x) = 1/(1-x) + x/(1 - x^2)^2 + x^4/(1 - x^3)^3 + x^9/(1 - x^4)^4 + x^16/(1 - x^5)^5 + x^25/(1 - x^6)^6 + x^36/(1 - x^7)^7 + x^49/(1 - x^8)^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A217667, A217668, A325997, A325998, A325999.

{a(n)=polcoeff(sum(m=0,n,(x+x^m +x*O(x^n))^m),n)}
for(n=0,100,print1(a(n),", "))

{a(n)=polcoeff(sum(m=0, n, x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+1)), n)}
for(n=0, 100, print1(a(n), ", "))

Generating functions.
(1) Sum_{n>=0} (x + x^n)^n.
(2) Sum_{n>=0} x^(n^2) / (1 - x^(n+1))^(n+1). - Paul D. Hanna, Jun 02 2019

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

Paul D. Hanna, Jul 17 2015

sign

			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 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..2050

Cf. A217668, A260147.

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), ", "))

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), ", "))

a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-n/d+1)*binomial(d, n/d-1))); \\ Seiichi Manyama, Feb 20 2023

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

A360712 Expansion of Sum_{k>0} (k * x * (1 + k*x^k))^k.

Original entry on oeis.org

1, 5, 27, 272, 3125, 46915, 823543, 16781312, 387421218, 10000078125, 285311670611, 8916102153177, 302875106592253, 11112006865911623, 437893890381640625, 18446744074783358976, 827240261886336764177, 39346408075327943829273

Offset: 1

Seiichi Manyama, Feb 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A217668, A327249, A338693, A359700, A360618.

a[n_] := DivisorSum[n, #^(#+n/#-1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x*(1+k*x^k))^k))

a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d, n/d-1));

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.

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

Paul D. Hanna, Jul 17 2015

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

			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 +...

Paul D. Hanna, Table of n, a(n) for n = 0..2050

Cf. A260116, A217668, A260147.

with(numtheory):
seq(add((-1)^(d-1)*binomial(n/d, d-1), d in divisors(n)), n = 1..70); # Peter Bala, Mar 02 2025

terms = 100; 1 + Sum[x^n*(1 - x^n)^n, {n, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)

{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), ", "))

{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), ", "))

{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), ", "))

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)

A326002 G.f.: Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n.

Original entry on oeis.org

1, 2, 5, 4, 11, 6, 22, 8, 29, 22, 41, 12, 89, 14, 71, 76, 109, 18, 214, 20, 196, 190, 155, 24, 573, 56, 209, 388, 519, 30, 877, 32, 809, 694, 341, 316, 2119, 38, 419, 1132, 2411, 42, 2045, 44, 2531, 2986, 599, 48, 6053, 106, 3011, 2500, 4759, 54, 4978, 4016, 6589, 3478, 929, 60, 21468, 62, 1055, 5524, 10713, 10076, 12046, 68, 13499, 6142, 18656, 72, 34474, 74, 1481, 29716, 20939, 5622, 28432, 80, 57921, 10000, 1805, 84, 84155, 42926, 1979, 12268, 41449, 90, 122339, 24116, 44759, 14974, 2351, 77616, 153969

Offset: 0

Paul D. Hanna, Jun 01 2019

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for some fixed integer k; here, k = 2 and p = 1, q = x, r = x.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 4*x^3 + 11*x^4 + 6*x^5 + 22*x^6 + 8*x^7 + 29*x^8 + 22*x^9 + 41*x^10 + 12*x^11 + 89*x^12 + 14*x^13 + 71*x^14 + 76*x^15 + 109*x^16 + 18*x^17 + 214*x^18 + 20*x^19 + 196*x^20 + ...
where we have the following series identity:
A(x) = 1 + 2*x*(1+x) + 3*x^2*(1+x^2)^2 + 4*x^3*(1+x^3)^3 + 5*x^4*(1+x^4)^4 + 6*x^5*(1+x^5)^5  + 7*x^6*(1+x^6)^6 + 8*x^7*(1+x^7)^7 + 9*x^8*(1+x^8)^8 + 10*x^9*(1+x^9)^9 + ...
is equal to
A(x) = 1/(1-x)^2 + 2*x^2/(1-x^2)^3 + 3*x^6/(1-x^3)^4 + 4*x^12/(1-x^4)^5 + 5*x^20/(1-x^5)^6 + 6*x^30/(1-x^6)^7 + 7*x^42/(1-x^7)^8 + 8*x^56/(1-x^8)^9 + ...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A217668 (k=1), A326003 (k=3), A326004 (k=4), A326005 (k=5).

N:= 100: # for a(0)..a(N)
S:= series(add((n+1)*x^n*(1+x^n)^n,n=0..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jun 03 2019

{a(n) = my(A = sum(m=0,n, (m+1) * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, (m+1) * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+2))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Generating functions.
(1) Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1) * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).

A326003 G.f.: Sum_{n>=0} (n+1)(n+2)/2 x^n * (1 + x^n)^n.

Original entry on oeis.org

1, 3, 9, 10, 27, 21, 64, 36, 105, 85, 171, 78, 359, 105, 372, 346, 573, 171, 1105, 210, 1116, 1009, 1134, 300, 3237, 456, 1743, 2386, 3375, 465, 5947, 528, 5529, 4885, 3537, 1926, 14917, 741, 4770, 9010, 16551, 903, 17963, 990, 19977, 22291, 8028, 1176, 49925, 1527, 23961, 24634, 41289, 1485, 48502, 27336, 58809, 37621, 15255, 1830, 184218, 1953, 18384, 59830, 106137, 77286, 121705, 2346, 140115, 78385, 159846, 2628, 346846, 2775, 30267, 293866

Offset: 0

Paul D. Hanna, Jun 01 2019

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),
for any fixed integer k; here, k = 3 and p = 1, q = x, r = x.

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 10*x^3 + 27*x^4 + 21*x^5 + 64*x^6 + 36*x^7 + 105*x^8 + 85*x^9 + 171*x^10 + 78*x^11 + 359*x^12 + 105*x^13 + 372*x^14 + 346*x^15 + 573*x^16 + 171*x^17 + 1105*x^18 + 210*x^19 + 1116*x^20 + ...
where we have the following series identity:
A(x) = 1 + 3*x*(1+x) + 6*x^2*(1+x^2)^2 + 10*x^3*(1+x^3)^3 + 15*x^4*(1+x^4)^4 + 21*x^5*(1+x^5)^5  + 28*x^6*(1+x^6)^6 + 36*x^7*(1+x^7)^7 + 45*x^8*(1+x^8)^8 + 55*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^3 + 3*x^2/(1-x^2)^4 + 6*x^6/(1-x^3)^5 + 10*x^12/(1-x^4)^6 + 15*x^20/(1-x^5)^7 + 21*x^30/(1-x^6)^8 + 28*x^42/(1-x^7)^9 + 36*x^56/(1-x^8)^10 +...

Paul D. Hanna, Table of n, a(n) for n = 0..2050

Cf. A217668 (k=1), A326002 (k=2), A326004 (k=4), A326005 (k=5).

{a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

{a(n) = my(A = sum(m=0,n, (m+1)*(m+2)/2 * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Generating functions.
(1) Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1)*(n+2)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+3).

A217667 G.f.: Sum_{n>=0} (x + x^(2*n))^n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 5, 1, 1, 5, 1, 4, 6, 1, 1, 7, 8, 1, 8, 1, 1, 19, 1, 5, 10, 1, 16, 11, 1, 1, 23, 22, 1, 13, 1, 1, 42, 21, 1, 20, 1, 37, 16, 1, 36, 17, 46, 1, 34, 1, 1, 130, 1, 1, 20, 1, 67, 56, 85, 7, 22, 79, 1, 23, 1, 121, 185, 1, 1, 25, 23, 106, 191, 1, 1

Offset: 0

Paul D. Hanna, Oct 10 2012

nonn

More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),
for any fixed integer k; here, k = 1 and y = 1, F = x, G = x^2.

			G.f.: A(x) = 1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 + x^7 + 5*x^8 + x^9 +...
where
A(x) = 1 + (x + x^2) + (x + x^4)^2 + (x + x^6)^3 + (x + x^8)^4 + (x + x^10)^5 +...
Also
A(x) = 1/(1-x) + x/(1 - x^3)^2 + x^4/(1 - x^5)^3 + x^9/(1 - x^7)^4 + x^16/(1 - x^9)^5 + x^25/(1 - x^11)^6 + x^36/(1 - x^13)^7 + x^49/(1 - x^15)^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A217668, A217669.

terms = 100; Sum[(x + x^(2*n))^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 16 2017 *)

{a(n)=polcoeff(sum(m=0,n,(x+x^(2*m) +x*O(x^n))^m),n)}
for(n=0,100,print1(a(n),", "))

Generating functions.
(1) Sum_{n>=0} (x + x^(2*n))^n.
(2) Sum_{n>=0} x^(n^2) / (1 - x^(2*n+1))^(n+1). - Paul D. Hanna, Jun 02 2019

A303506 G.f.: Sum_{n>=1} (-1)^(n-1) * x^(n^2)/(1 - x^n)^n.

Original entry on oeis.org

1, 1, 1, 0, 1, -1, 1, -2, 2, -3, 1, -1, 1, -5, 7, -7, 1, 3, 1, -12, 16, -9, 1, 1, 2, -11, 29, -32, 1, 28, 1, -49, 46, -15, 16, -18, 1, -17, 67, -67, 1, 53, 1, -140, 162, -21, 1, -103, 2, 103, 121, -244, 1, 55, 211, -305, 154, -27, 1, -17, 1, -29, 219, -486, 496, -73, 1, -592, 232, 766, 1, -931, 1, -35, 1278, -852, 211, -529, 1, 322, 327, -39, 1, -1654, 1821, -41, 379, -1492, 1, 750, 925, -1584

Offset: 1

Paul D. Hanna, Apr 25 2018

sign

			G.f.: A(x) = x + x^2 + x^3 + x^5 - x^6 + x^7 - 2*x^8 + 2*x^9 - 3*x^10 + x^11 - x^12 + x^13 - 5*x^14 + 7*x^15 - 7*x^16 + x^17 + 3*x^18 + ...
such that
A(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 - x^36/(1-x^6)^6 + x^49/(1-x^7)^7 - x^64/(1-x^8)^8 +...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A143862, A303340, A217668.

{a(n) = sumdiv(n,d, binomial(n/d-1, d-1) * (-1)^(d-1) )}
for(n=1,100, print1(a(n),", "))

a(n) = Sum_{d|n} binomial(n/d-1, d-1) * (-1)^(d-1) for n>=1.

cp's OEIS Frontend

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

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A217670 G.f.: Sum_{n>=0} x^n/(1 + x^n)^n.

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A217669 G.f.: Sum_{n>=0} (x + x^n)^n.

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

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A360712 Expansion of Sum_{k>0} (k * x * (1 + k*x^k))^k.

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

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A326003 G.f.: Sum_{n>=0} (n+1)(n+2)/2 x^n * (1 + x^n)^n.