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 1-3 of 3 results.

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

Views

Author

Paul D. Hanna, Sep 22 2022

Keywords

Comments

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).

Examples

			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) + ...

Links

Crossrefs

Cf. A291937, A356775, A357156, A357157.

Programs

  • PARI
    {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),", "))

Formula

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).

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).

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

Views

Author

Paul D. Hanna, Sep 22 2022

Keywords

Comments

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).

Examples

			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) + ...

Links

Crossrefs

Cf. A291937, A356774, A356775, A357157.

Programs

  • PARI
    {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),", "))

Formula

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^(4*n) * (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

Views

Author

Paul D. Hanna, Sep 22 2022

Keywords

Comments

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).

Examples

			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) + ...

Links

Crossrefs

Cf. A291937, A356774, A356775, A357156.

Programs

  • PARI
    {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),", "))

Formula

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).
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