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.

A357794 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).

Original entry on oeis.org

1, 3, 15, 114, 1086, 10824, 114382, 1252002, 14083275, 161810358, 1890774909, 22401092826, 268465408738, 3248818848876, 39643793276526, 487251937616006, 6026537732208078, 74954027622814455, 936840765257368687, 11761260253206563461, 148240496011414115676
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (y - x^n)^(n-1), which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 114*x^3 + 1086*x^4 + 10824*x^5 + 114382*x^6 + 1252002*x^7 + 14083275*x^8 + 161810358*x^9 + 1890774909*x^10 + ...

Links

Crossrefs

Cf. A357158, A357795, A357796.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 1 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^(n+1))^n * A(x)^(n+1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n*(n-2)) / ((1 - x^(n-1))^n * A(x)^(n-1)).

A357795 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).

Original entry on oeis.org

1, 4, 26, 300, 4134, 61696, 969660, 15837400, 266125823, 4571229248, 79904206064, 1416736880104, 25418030469904, 460600399886240, 8417980252615072, 154985730303047328, 2871904782258356719, 53519211809275995362, 1002383232008661189884, 18858606600633628740774
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (y - x^n)^(n-1), which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 300*x^3 + 4134*x^4 + 61696*x^5 + 969660*x^6 + 15837400*x^7 + 266125823*x^8 + 4571229248*x^9 + 79904206064*x^10 + ...

Links

Crossrefs

Cf. A357158, A357794, A357796.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-1, #A+1, n*(n+1)*(n+2)/3! * x^n * if(n==-2,0, (1 - x^(n+2) +x*O(x^#A) )^n) * Ser(A)^(n+2) ), #A-1) ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-1, #A+1, (-1)^(n-1) * n*(n-1)*(n-2)/3! * x^(n*(n-3)) * if(n==2,0, 1/(1 - x^(n-2) +x*O(x^#A) )^n) / Ser(A)^(n-2) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^n * (1 - x^(n+2))^n * A(x)^(n+2).
(2) 1 = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)/3! * x^(n*(n-3)) / ((1 - x^(n-2))^n * A(x)^(n-2)).

A357796 a(n) = coefficient of x^n in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^n * (1 - x^(n+3))^n * A(x)^(n+3).

Original entry on oeis.org

1, 5, 40, 635, 12095, 248245, 5381435, 121355095, 2817706420, 66909209195, 1617401484401, 39668321722180, 984661725380420, 24690230217076810, 624476169158179615, 15912858189842638180, 408139640637624168780, 10528308534373198776840, 272970775748658547320275
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (y - x^n)^(n-1), which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 5*x + 40*x^2 + 635*x^3 + 12095*x^4 + 248245*x^5 + 5381435*x^6 + 121355095*x^7 + 2817706420*x^8 + 66909209195*x^9 + 1617401484401*x^10 + ...

Links

Crossrefs

Cf. A357158, A357794, A357795.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-2, #A+2, n*(n+1)*(n+2)*(n+3)/4! * x^n * if(n==-3,0, (1 - x^(n+3) +x*O(x^#A) )^n) * Ser(A)^(n+3) ), #A-1) ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=-#A-2, #A+2, (-1)^n * n*(n-1)*(n-2)*(n-3)/4! * x^(n*(n-4)) * if(n==3,0, 1/(1 - x^(n-3) +x*O(x^#A) )^n) / Ser(A)^(n-3) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

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

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

Views

Author

Paul D. Hanna, Oct 03 2022

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A291937, A357158.

Programs

  • PARI
    {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),", "))

Formula

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).
Showing 1-4 of 4 results.

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