A260147 -id:A260147 - cp's OEIS frontend

A258936 G.f.: Sum_{n=-oo..+oo} x^n * (1 - 2^n*x^n)^n.

-1, -3, 7, -15, 89, -63, 121, -255, 3521, -13119, 18273, -4095, -40319, -16383, 425089, -2676735, 6141953, -262143, -22487551, -1048575, 173791233, -356171775, 176138241, -16777215, 2378907649, -5430575103, 3355336705, -38913703935, 164745740289, -1073741823, -770681831423, -4294967295, 4113638096897, -3796520402943, 1133869137921, -38542231207935, 87257121292289, -274877906943

Offset: 0

Paul D. Hanna, Nov 06 2015

sign

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n  =  0.
More generally, for all k we have the identity:
Sum_{n=-oo..+oo} x^n * (1 - k^n*x^n)^n = (-1) * Sum_{n=-oo..+oo} k*(k*x)^n * (1 - k*(k*x)^n)^n. - Paul D. Hanna, Dec 25 2015

			G.f.: A(x) = -1 - 3*x + 7*x^2 - 15*x^3 + 89*x^4 - 63*x^5 + 121*x^6 - 255*x^7 + 3521*x^8 - 13119*x^9 + 18273*x^10 - 4095*x^11 - 40319*x^12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A260147.

{a(n) = local(A=1); A = sum(k=-sqrtint(n)-1, n+1, x^k*(1 - 2^k*x^k + x*O(x^n))^k ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

G.f.: (-1) * Sum_{n=-oo..+oo} 2*(2*x)^n * (1 - 2*(2*x)^n)^n. - Paul D. Hanna, Dec 25 2015

It appears that for prime p >= 3, a(p) = 1 - 2^(p+1). - Peter Bala, Aug 06 2023

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

Paul D. Hanna, Jul 23 2015

nonn

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.

			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 +...

Cf. A260147, A217668, A260148, A363569, A363558, A363559, A363561.

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 *)

{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), ", "))

{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), ", "))

{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), ", "))

{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), ", "))

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).

A265266 G.f. A(x) satisfies: A(x) = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(2*n).

Original entry on oeis.org

1, 2, 5, 27, 143, 833, 5198, 33607, 223627, 1522249, 10546221, 74119591, 527150783, 3786896705, 27437431852, 200267244944, 1471209231873, 10869315344076, 80707738490984, 601977204069443, 4508156389422426, 33884634730883602, 255532279985062648, 1932864141175160374, 14660843479381675987, 111486308441258038306, 849773662058395948696, 6491244696415245552638, 49685280480631490670702, 381014689125058139363522, 2926949265189880054761750

Offset: 0

Paul D. Hanna, Jan 03 2016

nonn

Compare to: Sum_{n=-oo..+oo} x^n * (c - x^n)^n = 0 for fixed |c| > 0.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 27*x^3 + 143*x^4 + 833*x^5 + 5198*x^6 + 33607*x^7 + 223627*x^8 + 1522249*x^9 + 10546221*x^10 + ...
Let A = g.f. A(x) where A(x) = P(x) + N(x) then
P(x) = 1 + x*(A - x)^2 + x^2*(A - x^2)^4 + x^3*(A - x^3)^6 + x^4*(A - x^4)^8 + x^5*(A - x^5)^10 + x^6*(A - x^6)^12 + x^7*(A - x^7)^14 + x^8*(A - x^8)^16 + ...
N(x) = x/(1-x*A)^2 + x^6/(1-x^2*A)^4 + x^15/(1-x^3*A)^6 + x^28/(1-x^4*A)^8 + x^45/(1-x^5*A)^10 + x^66/(1-x^6*A)^12 + x^91/(1-x^7*A)^14 + ...
Explicitly,
P(x) = 1 + x + 3*x^2 + 20*x^3 + 117*x^4 + 708*x^5 + 4535*x^6 + 29801*x^7 + 200369*x^8 + 1373999*x^9 + 9570641*x^10 + 67539460*x^11 + 481899317*x^12 + ...
N(x) = x + 2*x^2 + 7*x^3 + 26*x^4 + 125*x^5 + 663*x^6 + 3806*x^7 + 23258*x^8 + 148250*x^9 + 975580*x^10 + 6580131*x^11 + 45251466*x^12 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A260147.

{a(n) = my(A=[1,1]); for(i=1,n, A=Vec( sum(n=-#A-1,#A+1, x^n*(Ser(A) - x^n)^(2*n) ) ) );A[n+1]}
for(n=0,40,print1(a(n),", "))

/* Quick print of terms 0..N (informal): */
N = 40; A=[1]; for(i=1,N, A=Vec( sum(n=-#A-1,#A+1, x^n*(Ser(A) - x^n)^(2*n) ) ) );A

The g.f. A(x) = Sum_{n>=0} a(n)*x^n also satisfies:
(1) A(x) = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(2*n).
(2) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - x^n*A(x))^(2*n).
(3) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + x^n*A(x))^(2*n).
(4) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (A(x^2) - x^n)^n.
(5) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (A(x^2) + x^n)^n.
(6) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n*A(x^2))^n.
(7) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n*A(x^2))^n.
a(n) ~ c * d^n / n^(3/2), where d = 8.078575206447883305059904... and c = 0.294232997886629805825... - Vaclav Kotesovec, Sep 03 2017

A266330 Triangle, read by rows, of the coefficients in the g.f.: Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n.

Original entry on oeis.org

-1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 0, 0, 0, 1, -1, 2, 0, -2, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 3, -1, 1, -3, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 4, 0, 0, 0, -4, 0, 0, 0, 1, -1, 0, -3, 0, 3, 0, 0, 0, 0, 0, 1, -1, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 6, -6, 1, -1, 6, 0, -6, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 1, -1, 0, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 8, 0, 4, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 9, -15, 0, 0, 0, 0, 15, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul D. Hanna, Dec 27 2015

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.
Note that the g.f.:
A(x,y) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n
may be written
A(x,y) = Sum_{n>=0} R(n,y) * x^n / y^(n+1)
such that row polynomials R(n,y) consist of square powers of y:
R(n,y) = Sum_{k=0..n+1} T(n,k) * y^(k^2).

			This triangle of coefficients T(n,k) begins:
  -1, 1;
  -1, 0, 1;
  -1, 1, -1, 1;
  -1, 0, 0, 0, 1;
  -1, 2, 0, -2, 0, 1;
  -1, 0, 0, 0, 0, 0, 1;
  -1, 3, -1, 1, -3, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 4, 0, 0, 0, -4, 0, 0, 0, 1;
  -1, 0, -3, 0, 3, 0, 0, 0, 0, 0, 1;
  -1, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 6, -6, 1, -1, 6, 0, -6, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 8, 0, 4, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  -1, 9, -15, 0, 0, 0, 0, 15, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 1; ...
in which the g.f. of column k > 0 is given by:
z^(k-1)*(1 - z^(k-1))^(k-1) + z^(k*(k+1))/(z^(k+1) - 1)^(k+1).
...
G.f.: A(x,y) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n may be written as
A(x,y) = Sum_{n>=0} R(n,y) * x^n / y^(n+1), where row polynomials R(n,y) consist of square powers of y:
R(n,y) = Sum_{k=0..n+1} T(n,k) * y^(k^2);
this triangle lists the coefficients of y^(k^2) in R(n,y), which begin:
  R(0,y) = y - 1;
  R(1,y) = y^4 - 1;
  R(2,y) = y^9 - y^4 + y - 1;
  R(3,y) = y^16 - 1;
  R(4,y) = y^25 - 2*y^9 + 2*y - 1;
  R(5,y) = y^36 - 1;
  R(6,y) = y^49 - 3*y^16 + y^9 - y^4 + 3*y - 1;
  R(7,y) = y^64 - 1;
  R(8,y) = y^81 - 4*y^25 + 4*y - 1;
  R(9,y) = y^100 + 3*y^16 - 3*y^4 - 1;
  R(10,y) = y^121 - 5*y^36 + 5*y - 1;
  R(11,y) = y^144 - 1;
  R(12,y) = y^169 - 6*y^49 + 6*y^25 - y^16 + y^9 - 6*y^4 + 6*y - 1;
  R(13,y) = y^196 - 1;
  R(14,y) = y^225 - 7*y^64 + 7*y - 1;
  R(15,y) = y^256 + 10*y^36 - 10*y^4 - 1;
  R(16,y) = y^289 - 8*y^81 - 4*y^25 + 4*y^9 + 8*y - 1;
  R(17,y) = y^324 - 1;
  R(18,y) = y^361 - 9*y^100 + 15*y^49 - 15*y^4 + 9*y - 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..5251, for rows 0..100 in flattened form of this triangle.

Cf. A217668, A260147 (y=-1).

/* Prints rows 0..50 of this triangle: */
{SUM=sum(n=-51,51,x^n*y^n*(y^n-x^n +O(x^51))^n); V=Vec(SUM);
T(n,k)=polcoeff(V[n+1]*y^(n+1) + y*O(y^((n+1)^2)), k^2)}
for(n=0,50,for(k=0,n+1,print1( T(n,k), ", "));print(""))

/* Quick print of row polynomials (informal): */
{SUM=sum(n=-51,51,x^n*y^n*(y^n-x^n +O(x^51))^n); V=Vec(SUM);
for(n=1,50,print("R("n-1",y) = "V[n]*y^n";")) }

/* Compare these sums (informal sanity check): */
Axy = sum(n=-16,16, x^n*y^n*(y^n-x^n +O(x^16))^n )
Axy = -(1/y)/(1-x/y) + sum(n=1,15, y^(n^2-1) * ( (x/y)^(n-1)*(1 - (x/y)^(n-1))^(n-1) + (x/y)^(n*(n+1)) / ((x/y)^(n+1) - 1)^(n+1) ) +O(x^16) )

G.f.: -(1/y)/(1-z) + (1/y) * Sum_{n>=1} y^(n^2) * ( z^(n-1)*(1 - z^(n-1))^(n-1) + z^(n*(n+1)) / (z^(n+1) - 1)^(n+1) ) = Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n, where z = x/y.
Row sums are all zeros.
Row sums of absolute values of terms yield 2 * A217668(n) for row n>=0.
Sum_{k=0..2*n+1} (-1)^k * T(2*n,k) = (-2) * A260147(n) for n>=0.
Sum_{k=0..2*n+2} (-1)^k * T(2*n+1,k) = 0 for n>=0.
Sum_{k=0..2*n+1} I^(k^2) * T(2*n,k) = (I-1) * A260147(n) for n>=0, where I^2 = -1.
Sum_{k=0..2*n+2} I^(k^2) * T(2*n+1,k) = 0 for n>=0, where I^2 = -1.

A268298 G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^(3*n).

Original entry on oeis.org

1, 1, -3, 1, -12, -9, -8, -20, -59, 1, -43, -54, -101, -77, -89, 127, -307, -135, -26, -170, 73, 85, 199, -252, -888, 1066, 924, 1, 1177, -405, 2970, -464, 1009, -164, 5577, 10396, -2978, -665, 10869, -1286, 14576, -819, 15499, -902, 19934, 17551, 32546, -1080, -51905, 53089, 74231, -24309, 55317, -1377, -80, 42439, 103857, -75581, 117016, -1710

Offset: 0

Paul D. Hanna, Mar 29 2016

sign

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

			G.f.: A(x) = 1 + x - 3*x^2 + x^3 - 12*x^4 - 9*x^5 - 8*x^6 - 20*x^7 - 59*x^8 + x^9 - 43*x^10 - 54*x^11 - 101*x^12 - 77*x^13 - 89*x^14 + 127*x^15 +...

Paul D. Hanna, Table of n, a(n) for n = 0..2200

Cf. A260147.

{a(n) = local(A=1); A = sum(k=-n-1, n+1, x^k*(1-x^k + x*O(x^n) )^(3*k) ); polcoeff(A, n)}
for(n=0, 60, print1(a(n), ", "))

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

For n>0, a(n) = 1 iff n = 3^k for k>=0 (conjecture).

cp's OEIS Frontend

A258936 G.f.: Sum_{n=-oo..+oo} x^n * (1 - 2^n*x^n)^n.

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A260361 G.f.: Sum_{n=-oo..+oo} x^n * (1 + x^n)^n, an even function.

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A265266 G.f. A(x) satisfies: A(x) = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(2*n).

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A266330 Triangle, read by rows, of the coefficients in the g.f.: Sum_{n=-oo..+oo} x^n * y^n * (y^n - x^n)^n.

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A268298 G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^(3*n).

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