A291937 -id:A291937 - cp's OEIS frontend

A292184 a(n) = A291937(3^n).

2, 4, 28, 460, 10774, 80195104, 2894790054826, 122274810705200924689300, 17750307143185064814011639706060016204

Offset: 0

Paul D. Hanna, Oct 05 2017

G.f. of A291937 equals: Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n.

It is surprising that these terms seem to be all positive and increasing so rapidly.

			These terms are located at positions 3^n of sequence A291937, whose g.f. begins:
G(x) = 1 + (2)*x + (4)*x^3 - 3*x^4 + 6*x^5 - 3*x^6 + 8*x^7 - 15*x^8 + (28)*x^9 - 24*x^10 + 12*x^11 + 14*x^13 - 48*x^14 + 96*x^15 - 95*x^16 + 18*x^17 + 55*x^18 + 20*x^19 - 180*x^20 + 232*x^21 - 120*x^22 + 24*x^23 - 35*x^24 + 76*x^25 - 168*x^26 + (460)*x^27 - 580*x^28 + 30*x^29 + 515*x^30 +...
such that
G(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^n.
Also,
G(x) = Sum_{n=-oo..+oo} n^2 * x^(2*n) * (1 - x^n)^(n-1).

Cf. A291937.

A293129 L.g.f.: Sum_{n=-oo..+oo} (x - x^(2n-1))^(2n-1) / (2*n-1).

Original entry on oeis.org

1, 4, 1, 15, 1, 12, 40, 16, 1, 77, 92, 24, 101, 28, 204, 373, 1, 36, 667, 40, 575, 689, 826, 48, 393, 1582, 1379, 1937, 590, 60, 6101, 64, 1, 5227, 3129, 9515, 1826, 76, 4390, 12404, 11341, 84, 18361, 88, 5875, 46320, 7844, 96, 1553, 33133, 38886, 50883, 25741, 108, 25507, 44993, 82265, 91449, 15835, 120, 150162, 124, 19376, 390653, 1, 104015, 29394, 136, 242217, 249506, 507789, 144, 210831, 148, 33079, 647187, 593029, 711482, 47101, 160

Offset: 1

Paul D. Hanna, Oct 11 2017

nonn

Compare l.g.f. to: Sum_{n=-oo..+oo, n<>0} (x - x^n)^n / n  =  -log(1-x).
Here l.g.f. L(x) = Sum_{n>=1} a(n) * x^(2*n-1) / (2*n-1).
a(2^n + 1) = 1 for n >= 1 (conjecture).

			L.g.f.: L(x) = x + 4*x^3/3 + x^5/5 + 15*x^7/7 + x^9/9 + 12*x^11/11 + 40*x^13/13 + 16*x^15/15 + x^17/17 + 77*x^19/19 + 92*x^21/21 + 24*x^23/23 + 101*x^25/25 + 28*x^27/27 + 204*x^29/29 + 373*x^31/31 + x^33/33 + 36*x^35/35 + 667*x^37/37 + 40*x^39/39 + 575*x^41/41 + 689*x^43/43 + 826*x^45/45 + 48*x^47/47 + 393*x^49/49 + 1582*x^51/51 + 1379*x^53/53 + 1937*x^55/55 + 590*x^57/57 + 60*x^59/59 +...
such that L(x) = Sum_{n=-oo..+oo} (x - x^(2*n-1))^(2*n-1) / (2*n-1).
The coefficient of x^(2^n+1)/(2^n+1) in L(x) for n>=1 begins:
[4, 1, 1, 1, 1, 1, 1, 1, 1, ...],
and it appears that a(k) = 1 only at k = 1 and k = 2^n + 1 (n>=1).
We may write L(x) = P(x) + Q(x) where
P(x) = (x - x) + (x - x^3)^3/3 + (x - x^5)^5/5 + (x - x^7)^7/7 + (x - x^9)^9/9 + (x - x^11)^11/11 + (x - x^13)^13/13 + (x - x^15)^15/15 + (x - x^17)^17/17 + (x - x^19)^19/19 + (x - x^21)^21/21 +...+ (x - x^(2*n-1))^(2*n-1)/(2*n-1) +...
Q(x) = x/(1 - x^2) + x^9/(3*(1 - x^4)^3) + x^25/(5*(1 - x^6)^5) + x^49/(7*(1 - x^8)^7) + x^81/(9*(1 - x^10)^9) + x^121/(11*(1 - x^12)^11) + x^169/(13*(1 - x^14)^13) +...+ x^((2*n-1)^2) / ((2*n-1)*(1 - x^(2*n))^(2*n-1)) +...
Explicitly,
P(x) = x^3/3 - 4*x^5/5 + 8*x^7/7 - 11*x^9/9 + x^11/11 + 14*x^13/13 + x^15/15 - 50*x^17/17 + 58*x^19/19 + x^21/21 + x^23/23 - 54*x^25/25 + x^27/27 - 28*x^29/29 + 311*x^31/31 - 340*x^33/33 + x^35/35 + 75*x^37/37 + x^39/39 - 81*x^41/41 + 345*x^43/43 - 44*x^45/45 + x^47/47 - 1427*x^49/49 + 1531*x^51/51 - 52*x^53/53 + 496*x^55/55 - 1253*x^57/57 + x^59/59 + 1343*x^61/61 + x^63/63 - 2924*x^65/65 +...
Q(x) = x + 3*x^3/3 + 5*x^5/5 + 7*x^7/7 + 12*x^9/9 + 11*x^11/11 + 26*x^13/13 + 15*x^15/15 + 51*x^17/17 + 19*x^19/19 + 91*x^21/21 + 23*x^23/23 + 155*x^25/25 + 27*x^27/27 + 232*x^29/29 + 62*x^31/31 + 341*x^33/33 + 35*x^35/35 + 592*x^37/37 + 39*x^39/39 + 656*x^41/41 + 344*x^43/43 + 870*x^45/45 + 47*x^47/47 + 1820*x^49/49 + 51*x^51/51 + 1431*x^53/53 + 1441*x^55/55 + 1843*x^57/57 + 59*x^59/59 + 4758*x^61/61 + 63*x^63/63 + 2925*x^65/65 +...
The coefficient of x^(2^n+1)/(2^n+1) in P(x) for n>=1 begins:
[1, -4, -11, -50, -340, -2924, -169032, -33445208, -21619038032, 1 - A293599(n), ...].
The coefficient of x^(2^n+1)/(2^n+1) in Q(x) for n>=1 begins:
[3, 5, 12, 51, 341, 2925, 169033, 33445209, 21619038033, ..., A293599(n), ...].

Paul D. Hanna, Table of n, a(n) for n = 1..2050

Cf. A293597 (P(x)), A293598 (Q(x)), A293599, A291937.

{a(n) = my(P,Q,Ox = O(x^(2*n+1)));
P = sum(m=1,n+1, (x - x^(2*m-1) +Ox)^(2*m-1) / (2*m-1) );
Q = sum(m=1,sqrtint(n+1), x^((2*m-1)^2) / ( (2*m-1) * (1 - x^(2*m) +Ox)^(2*m-1) ) );
(2*n-1)*polcoeff(P + Q, 2*n-1)}
for(n=1,80,print1(a(n),", "))

L.g.f.: Sum_{n=-oo..+oo} (x + x^(2*n-1))^(2*n-1) / (2*n-1) - note the plus sign.
L.g.f.: -log(1-x) - Sum_{n=-oo..+oo, n<>0} (x - x^(2*n))^(2*n) / (2*n).
L.g.f.: L(x) = P(x) + Q(x) where
P(x) = Sum_{n>=1} (x - x^(2*n-1))^(2*n-1) / (2*n-1),
Q(x) = Sum_{n>=1} x^((2*n-1)^2) / ( (2*n-1) * (1 - x^(2*n))^(2*n-1) ).

A356774 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-2).

Original entry on oeis.org

1, 4, 7, 11, 16, 17, 29, 21, 46, 21, 67, 22, 92, 1, 151, -23, 154, 22, 191, -118, 407, -175, 277, 23, 326, -363, 946, -643, 436, 282, 497, -1199, 1948, -1019, 701, -47, 704, -1519, 3641, -3127, 862, 1759, 947, -5301, 7036, -2943, 1129, -1187, 1226, -2149, 10252

Offset: 1

Paul D. Hanna, Sep 22 2022

sign

Related identities:
(I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).
(I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).
(I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).
(I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).
(I.5) 0 = Sum_{n=-oo..+oo} binomial(n+k-1, k) * x^(k*n) * (1 - x^n)^(n-1) for fixed positive integer k.
(I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).
(I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).
(I.8) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)/3! * x^(n*(n+2)) / (1 - x^(n+2))^(n+3).
(I.9) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)*(n+3)/4! * x^(n*(n+3)) / (1 - x^(n+3))^(n+4).
(I.10) 0 = Sum_{n=-oo..+oo} (-1)^n * binomial(n+k-1, k) * x^(n*(n+k-1)) / (1 - x^(n+k-1))^(n+k) for fixed positive integer k.
(I.11) 0 = Sum_{n=-oo..+oo} n*(n-1)/2 * x^n * (1 - x^n)^(n-2).
(I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^n)^(n+2).

			G.f.: A(x) = x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 17*x^6 + 29*x^7 + 21*x^8 + 46*x^9 + 21*x^10 + 67*x^11 + 22*x^12 + 92*x^13 + x^14 + 151*x^15 + ...
where
A(x) = ... - 3*x^(-3)*(1 - x^(-3))^(-5) - 2*x^(-2)*(1 - x^(-2))^(-4) - x^(-1)*(1 - x^(-1))^(-3) + 0 + x/(1-x) + 2*x^2 + 3*x^3*(1 - x^3) + 4*x^4*(1 - x^4)^2 + 5*x^5*(1 - x^5)^3 + ... + n*x^n*(1 - x^n)^(n-2) + ...

Paul D. Hanna, Table of n, a(n) for n = 1..2050

Cf. A291937, A356775, A357156, A357157.

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

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-2).
(2) A(x) = Sum_{n=-oo..+oo} n * x^(2*n) * (1 - x^n)^(n-2).
(3.a) A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^n)^(n-2).
(3.b) A(x) = Sum_{n=-oo..+oo} n^2 * x^n * (1 - x^n)^(n-2).
(4) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n * x^(n^2) / (1 - x^n)^(n+2).
(5) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n * x^(n*(n+1)) / (1 - x^n)^(n+2).
(6.a) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n*(n+1)) / (1 - x^n)^(n+2).
(6.b) A(x) = Sum_{n=-oo..+oo} (-1)^n * n^2 * x^(n*(n+1)) / (1 - x^n)^(n+2).

A356775 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n(n+1)/2 x^(2n) (1 - x^n)^(n-2).

Original entry on oeis.org

1, 1, 5, 1, 11, 1, 21, -8, 36, 1, 22, 1, 85, -89, 137, 1, -23, 1, 302, -349, 287, 1, 23, -24, 456, -944, 1177, 1, -903, 1, 2113, -2078, 970, -559, 709, 1, 1331, -4003, 4293, 1, -3323, 1, 9153, -10694, 2301, 1, 5869, -48, -4774, -11474, 20294, 1, -7334, -14783

Offset: 2

Paul D. Hanna, Sep 22 2022

sign

Related identities:
(I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).
(I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).
(I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).
(I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).
(I.5) 0 = Sum_{n=-oo..+oo} binomial(n+k-1, k) * x^(k*n) * (1 - x^n)^(n-1) for fixed positive integer k.
(I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).
(I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).
(I.8) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)/3! * x^(n*(n+2)) / (1 - x^(n+2))^(n+3).
(I.9) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)*(n+3)/4! * x^(n*(n+3)) / (1 - x^(n+3))^(n+4).
(I.10) 0 = Sum_{n=-oo..+oo} (-1)^n * binomial(n+k-1, k) * x^(n*(n+k-1)) / (1 - x^(n+k-1))^(n+k) for fixed positive integer k.
(I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)/6 * x^(2*n) * (1 - x^n)^(n-2).
(I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n-1)*n*(n+1)/6 * x^(n^2) / (1 - x^n)^(n+2).

			G.f.: A(x) = x^2 + x^3 + 5*x^4 + x^5 + 11*x^6 + x^7 + 21*x^8 - 8*x^9 + 36*x^10 + x^11 + 22*x^12 + x^13 + 85*x^14 - 89*x^15 + 137*x^16 + ...
where
A(x) = ... + 3*x^(-6)*(1 - x^(-3))^(-5) + 1*x^(-4)*(1 - x^(-2))^(-4) + 0*x^(-2) + 0 + 1*x^2/(1-x) + 3*x^4 + 6*x^6*(1 - x^3) + 10*x^8*(1 - x^4)^2 + 15*x^10*(1 - x^5)^3 + ... + n*(n+1)/2 * x^(2*n)*(1 - x^n)^(n-2) + ...

Paul D. Hanna, Table of n, a(n) for n = 2..2050

Cf. A291937, A356774, A357156, A357157.

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

G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies:
(1) A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-2).
(2) A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(3*n) * (1 - x^n)^(n-2).
(3) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(2*n) * (1 - x^n)^(n-2).
(4) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n*(n-1)) / (1 - x^n)^(n+2).
(5) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n^2) / (1 - x^n)^(n+2).
(6) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)/6 * x^(n^2) / (1 - x^n)^(n+2).

A357156 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n(n+1)(n+2)/6 * x^(3n) (1 - x^n)^(n-2).

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 16, 1, 1, 22, 1, 1, 71, -63, 1, 127, 1, -158, 211, 1, 1, -117, 176, 1, 496, -923, 1, 1277, 1, -1727, 1002, 1, 1681, -2021, 1, 1, 1821, -1027, 1, 912, 1, -7721, 11146, 1, 1, -12571, 736, 15401, 4846, -17016, 1, -6389, 27457, -20956, 7316, 1, 1, -6486, 1, 1, 22177

Offset: 3

Paul D. Hanna, Sep 22 2022

sign

Related identities:
(I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).
(I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).
(I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).
(I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).
(I.5) 0 = Sum_{n=-oo..+oo} binomial(n+k-1, k) * x^(k*n) * (1 - x^n)^(n-1) for fixed positive integer k.
(I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).
(I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).
(I.8) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)/3! * x^(n*(n+2)) / (1 - x^(n+2))^(n+3).
(I.9) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)*(n+3)/4! * x^(n*(n+3)) / (1 - x^(n+3))^(n+4).
(I.10) 0 = Sum_{n=-oo..+oo} (-1)^n * binomial(n+k-1, k) * x^(n*(n+k-1)) / (1 - x^(n+k-1))^(n+k) for fixed positive integer k.
(I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)*(n+2)/24 * x^(3*n) * (1 - x^n)^(n-2).
(I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n+1)*n*(n-1)*(n-2)/24 * x^(n*(n-1)) / (1 - x^n)^(n+2).

			G.f.: A(x) = x^3 + x^4 + x^5 + 6*x^6 + x^7 + x^8 + 16*x^9 + x^10 + x^11 + 22*x^12 + x^13 + x^14 + 71*x^15 - 63*x^16 + x^17 + 127*x^18 + ...
where
A(x) = ... - 4*x^(-12)*(1 - x^(-4))^(-6) - 1*x^(-9)*(1 - x^(-3))^(-5) + 0*x^(-6) + 0*x^(-3) + 0 + 1*x^3/(1-x) + 4*x^6 + 10*x^9*(1 - x^3) + 20*x^12*(1 - x^4)^2 + 35*x^15*(1 - x^5)^3 + ... + n*(n+1)*(n+2)/6 * x^(3*n)*(1 - x^n)^(n-2) + ...

Paul D. Hanna, Table of n, a(n) for n = 3..2050

Cf. A291937, A356774, A356775, A357157.

{a(n) = my(A = sum(m=-n-1,n+1, if(m==0,0, m*(m+1)*(m+2)/6 * x^(3*m) * (1 - x^m +x*O(x^n))^(m-2) )) );
polcoeff(A,n)}
for(n=3,100,print1(a(n),", "))

G.f. A(x) = Sum_{n>=3} a(n)*x^n satisfies:
(1) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(3*n) * (1 - x^n)^(n-2).
(2) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(4*n) * (1 - x^n)^(n-2).
(3) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(3*n) * (1 - x^n)^(n-2).
(4) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)/6 * x^(n*(n-2)) / (1 - x^n)^(n+2).
(5) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)/6 * x^(n*(n-1)) / (1 - x^n)^(n+2).
(6) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-1)) / (1 - x^n)^(n+2).

A357157 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n(n+1)(n+2)(n+3)/24 x^(4n) (1 - x^n)^(n-2).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 1, 22, 1, 1, -19, 57, 1, 1, 1, 22, 1, 1, 1, 303, -349, 1, 1, 463, 1, -593, 1, 793, 1, 1, -2204, 2584, 1, 1, 1, -2287, 1, 3082, 1, 3004, -8084, 1, 1, 14465, -3674, -14299, 1, 6189, 1, 22276, -24023, -2056, 1, 1, 1, 18714, 1, 1, -34985, 24305, -60059, 87517, 1, 20350

Offset: 4

Paul D. Hanna, Sep 22 2022

sign

Related identities:
(I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).
(I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).
(I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).
(I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).
(I.5) 0 = Sum_{n=-oo..+oo} binomial(n+k-1, k) * x^(k*n) * (1 - x^n)^(n-1) for fixed positive integer k.
(I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).
(I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).
(I.8) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)/3! * x^(n*(n+2)) / (1 - x^(n+2))^(n+3).
(I.9) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)*(n+3)/4! * x^(n*(n+3)) / (1 - x^(n+3))^(n+4).
(I.10) 0 = Sum_{n=-oo..+oo} (-1)^n * binomial(n+k-1, k) * x^(n*(n+k-1)) / (1 - x^(n+k-1))^(n+k) for fixed positive integer k.
(I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)*(n+2)*(n+3)/120 * x^(4*n) * (1 - x^n)^(n-2).
(I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n+1)*n*(n-1)*(n-2)*(n-3)/120 * x^(n*(n-2)) / (1 - x^n)^(n+2).

			G.f.: A(x) = x^4 + x^5 + x^6 + x^7 + 7*x^8 + x^9 + x^10 + x^11 + 22*x^12 + x^13 + x^14 - 19*x^15 + 57*x^16 + x^17 + x^18 + x^19 + 22*x^20 + ...
where
A(x) = ... + 5*x^(-20)*(1 - x^(-5))^(-7) + 1*x^(-16)*(1 - x^(-4))^(-6) + 0*x^(-12) + 0*x^(-8) + 0*x^(-4) + 0 + 1*x^4/(1-x) + 5*x^8 + 15*x^12*(1 - x^3) + 35*x^16*(1 - x^4)^2 + 70*x^20*(1 - x^5)^3 + ... + n*(n+1)*(n+2)*(n+3)/24 * x^(4*n)*(1 - x^n)^(n-2) + ...

Paul D. Hanna, Table of n, a(n) for n = 4..2050

Cf. A291937, A356774, A356775, A357156.

{a(n) = my(A = sum(m=-n-1,n+1, if(m==0,0, m*(m+1)*(m+2)*(m+3)/24 * x^(4*m) * (1 - x^m +x*O(x^n))^(m-2) )) );
polcoeff(A,n)}
for(n=4,100,print1(a(n),", "))

G.f. A(x) = Sum_{n>=4} a(n)*x^n satisfies:
(1) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(4*n) * (1 - x^n)^(n-2).
(2) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(5*n) * (1 - x^n)^(n-2).
(3) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)*(n+4)/120 * x^(4*n) * (1 - x^n)^(n-2).
(4) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-3)) / (1 - x^n)^(n+2).
(5) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-2)) / (1 - x^n)^(n+2).
(6) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)*(n-3)*(n-4)/120 * x^(n*(n-2)) / (1 - x^n)^(n+2).

A292177 G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n(n+1)/6 x^(n+K))^n.

Original entry on oeis.org

1, 0, 4, 0, 5, 0, 20, -24, 35, 0, 0, 0, 84, -160, 200, 0, -150, 0, 460, -560, 286, 0, 140, -200, 455, -1440, 2100, 0, -2180, 0, 3840, -3080, 969, -2240, 2730, 0, 1330, -5824, 5320, 0, -4235, 0, 16874, -21840, 2300, 0, 18440, -784, -20175, -16320, 37310, 0, -945, -42240, 49560, -25080, 4495, 0, 7560, 0, 5456, -50400, 102528, -120120, 40810, 0, 135660, -52624, -221690, 0, 278256, 0, 9139, -364000, 232750, -99792, 211120, 0, -106680, -100440, 12341, 0, 537992, -628320, 14190, -129920, 563420, 0, -195015, -480480, 591100, -168640, 18424, -1240320, 2138640, 0, -925120, -268224, -803250, 0

Offset: 2

Paul D. Hanna, Sep 10 2017

sign

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

			G.f.: A(x) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 35*x^10 + 84*x^14 - 160*x^15 + 200*x^16 - 150*x^18 + 460*x^20 - 560*x^21 + 286*x^22 + 140*x^24 - 200*x^25 + 455*x^26 - 1440*x^27 + 2100*x^28 - 2180*x^30 + 3840*x^32 - 3080*x^33 + 969*x^34 - 2240*x^35 + 2730*x^36 + 1330*x^38 - 5824*x^39 + 5320*x^40 +...
such that the g.f. equals the limit of the sum, as K tends to infinity,
S(K) = Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n*(n+1)/6 * x^(n+K))^n.
Illustration of the limit.
S(1) = x^2 + 1/3*x^3 + 112/27*x^4 + 113/81*x^5 + 467/81*x^6 - 938/729*x^7 +...
S(2) = x^2 + 13/3*x^4 + 175/27*x^6 + 1550/81*x^8 - 24*x^9 + 2777/81*x^10 +...
S(3) = x^2 + 4*x^4 + 1/3*x^5 + 5*x^6 + 4/3*x^7 + 544/27*x^8 - 77/3*x^9 +...
S(4) = x^2 + 4*x^4 + 16/3*x^6 + 64/3*x^8 - 24*x^9 + 904/27*x^10 +...
S(5) = x^2 + 4*x^4 + 5*x^6 + 1/3*x^7 + 20*x^8 - 68/3*x^9 + 35*x^10 +...
S(6) = x^2 + 4*x^4 + 5*x^6 + 61/3*x^8 - 24*x^9 + 109/3*x^10 - 5/3*x^12 +...
S(7) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 71/3*x^9 + 35*x^10 + 4/3*x^11 +...
S(8) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 106/3*x^10 + 4/3*x^12 +...
S(9) = x^2 + 4*x^4 + 5*x^6 + 20*x^8 - 24*x^9 + 35*x^10 + 1/3*x^11 +...
...
At powers of 2, a(2^n) begins:
[1, 4, 20, 200, 3840, 102528, 8437440, 5275875200, 5635011683840, 2075681844543566848, 671078483184128826885120, ...].

Paul D. Hanna, Table of n, a(n) for n = 2..1002

Cf. A291937.

{a(n) = my(A=1,K=n); A = sum(m=-sqrtint(2*n+9), 2*n+1, x^(m-K) * (1 - x^m +m*(m+1)/6*x^(m+K) + O(x^(2*n+2)) )^m  ); polcoeff(A, n)}
for(n=2, 80, print1(a(n), ", "))

G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} -(-1)^n * x^(n^2-n-K) / (1 - x^n + n*(n-1)/6 * x^K)^n.

a(p) = 0 for odd prime p (conjecture).

A357159 a(n) = coefficient of x^n in the power series A(x) such that: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n, starting with a(0) = -1.

Original entry on oeis.org

-1, -2, -4, -8, -8, -6, 40, 132, 400, 504, 76, -4960, -18528, -56998, -94176, -58896, 617216, 2911128, 9741760, 19739472, 21657312, -75073186, -483271024, -1800924184, -4274295720, -6374947674, 7150661892, 81254492928, 345397065128, 937137978804, 1717431001440

Offset: 0

Paul D. Hanna, Oct 03 2022

sign

Related identity: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1), which holds when 0 < |x| < 1.

Note that Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n is to be taken as the sum of two infinite series, P(x) + Q(x), where P(x) = Sum_{n=-oo..-1} n * x^n * (1 - x^n)^(n-1) * A(x)^n and Q(x) = Sum_{n=+1..+oo} n * x^n * (1 - x^n)^(n-1) * A(x)^n. The g.f. A(x) of this sequence satisfies the condition that P(x) + Q(x) = 0. The series Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n converges to zero when 0 < |x| < r where r < 1 is the radius of convergence of g.f. A(x). Upon reversing the sign of the index n, and so taking the same sum in reverse order from +oo to -oo, we obtain the equivalent series Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) / ((1 - x^n)^(n+1) * A(x)^n), the convergence of which is more clearly seen to hold when 0 < |x| < r < 1.

			G.f.: A(x) = -1 - 2*x - 4*x^2 - 8*x^3 - 8*x^4 - 6*x^5 + 40*x^6 + 132*x^7 + 400*x^8 + 504*x^9 + 76*x^10 - 4960*x^11 - 18528*x^12 - 56998*x^13 - 94176*x^14 - 58896*x^15 + 617216*x^16 + ...
such that
0 = ... - 3*(x*A(x))^(-3)/(1 - x^(-3))^4 - 2*(x*A(x))^(-2)/(1 - x^(-2))^3 - (x*A(x))^(-1)/(1 - x^(-1))^2 + 0 + x*A(x) + 2*(x*A(x))^2*(1 - x^2) + 3*(x*A(x))^3*(1 - x^3)^2 + 4*(x*A(x))^4*(1 - x^4)^3 + 5*(x*A(x))^5*(1 - x^5)^4 + ... + n*(x*A(x))^n*(1 - x^n)^(n-1) + ...
SPECIFIC VALUES.
A(1/4) = -1.8892616570712410815999763792198265088...
A(1/5) = -1.6334109911560757412636074394753603214...
A(1/6) = -1.4868349923582400870800926746579742411...
We can illustrate the sum in the definition at x = 1/4.
The sum
0 = Sum_{n=-oo..+oo, n<>0} n * 1/4^n * (1 - 1/4^n)^(n-1) * A(1/4)^n
simplifies somewhat to
0 = Sum_{n=-oo..+oo, n<>0} n * (4^n - 1)^(n-1) * A(1/4)^n / 4^(n^2),
which can be split up into parts P and Q.
Let P denote the sum from -oo to -1, which can be written as
P = Sum_{n>1} (-1)^n * n * 4^n / ((4^n - 1)^(n+1) * A(1/4)^n),
and let Q denote the sum from +1 to +oo:
Q = Sum_{n>1} n * (4^n - 1)^(n-1) * A(1/4)^n / 4^(n^2).
Substituting A(1/4) = -1.8892616570712410815999763792198265088... yields
P = 0.237905890404564510234837963872429856... and
Q = -0.237905890404564510234837963872429856...
so that P + Q = 0.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A291937, A357158.

{a(n) = my(A=[-1]); for(i=1,n, A=concat(A,0);
A[#A] = -polcoeff( sum(n=-#A,#A, if(n==0,0, n * x^n * (1 - x^n +x*O(x^#A) )^(n-1) * Ser(A)^n )),#A)/2 );A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n, with a(0) = -1, satisfies the following relations.
(1) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n.
(2) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n/A(x))^(n-1).
(3) 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (A(x) - x^n)^(n-1).
(4) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) / ( (1 - x^n)^(n+1) * A(x)^n ).
(5) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) * A(x)^n / (A(x) - x^n)^(n+1).
(6) 0 = Sum_{n=-oo..+oo, n<>0} (-1)^n * n * x^(n^2) * A(x)^n / (1 - x^n*A(x))^(n+1).

cp's OEIS Frontend

A292184 a(n) = A291937(3^n).

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A293129 L.g.f.: Sum_{n=-oo..+oo} (x - x^(2*n-1))^(2*n-1) / (2*n-1).

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A356774 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-2).

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A356775 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-2).

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A357156 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(3*n) * (1 - x^n)^(n-2).

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A357157 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(4*n) * (1 - x^n)^(n-2).

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A292177 G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n*(n+1)/6 * x^(n+K))^n.

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A357159 a(n) = coefficient of x^n in the power series A(x) such that: 0 = Sum_{n=-oo..+oo, n<>0} n * x^n * (1 - x^n)^(n-1) * A(x)^n, starting with a(0) = -1.

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A293129 L.g.f.: Sum_{n=-oo..+oo} (x - x^(2n-1))^(2n-1) / (2*n-1).

A356775 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n(n+1)/2 x^(2n) (1 - x^n)^(n-2).

A357156 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n(n+1)(n+2)/6 * x^(3n) (1 - x^n)^(n-2).

A357157 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n(n+1)(n+2)(n+3)/24 x^(4n) (1 - x^n)^(n-2).

A292177 G.f.: Limit_{K->oo} Sum_{n=-oo..+oo} x^(n-K) * (1 - x^n + n(n+1)/6 x^(n+K))^n.