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 1-4 of 4 results.

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).

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

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)/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).

Examples

			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) + ...

Links

Crossrefs

Cf. A291937, A356774, A357156, A357157.

Programs

  • PARI
    {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),", "))

Formula

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^(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).

A357406 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^(2*n+2) * (1 - x^n)^(n+1).

Original entry on oeis.org

1, 0, -1, 0, 3, -8, 9, 0, -10, 0, 24, -24, 0, 0, 15, 0, 9, -80, 90, 0, -43, 0, 57, -80, 13, 0, 175, -200, 15, -120, 313, 0, -346, 0, 450, -168, 19, -744, 830, 0, 21, -224, -287, 0, 405, 0, 1014, -1968, 25, 0, 2813, -784, -2448, -360, 1575, 0, 2765, -3520, 450, -440, 31
Offset: 0

Views

Author

Paul D. Hanna, Sep 27 2022

Keywords

Examples

			G.f.: A(x) = 1 - x^2 + 3*x^4 - 8*x^5 + 9*x^6 - 10*x^8 + 24*x^10 - 24*x^11 + 15*x^14 + 9*x^16 - 80*x^17 + 90*x^18 - 43*x^20 + 57*x^22 - 80*x^23 + 13*x^24 + ...
Related series.
x/A(x) = x + x^3 - 2*x^5 + 8*x^6 - 14*x^7 + 16*x^8 - 7*x^9 - 24*x^10 + 103*x^11 - 232*x^12 + 334*x^13 - 256*x^14 - 211*x^15 + 1400*x^16 + ... + A357401(n)*x^n + ...

Links

Crossrefs

Cf. A357401, A356774.

Programs

  • PARI
    {a(n) = my(A = sum(m=-n\2-1,n\2+1, m * x^(2*m+2) * (1 - x^m +x*O(x^n) )^(m+1)) ); polcoeff(A,n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = Sum_{n=-oo..+oo} n * x^(2*n+2) * (1 - x^n)^(n+1),
(2) A(x) = -Sum_{n=-oo..+oo, n<>0} n * (-1)^n * x^((n-1)*(n-2)) / (1 - x^n)^(n-1).
Showing 1-4 of 4 results.

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