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.

A359721 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 3, 10, 37, 127, 460, 1710, 6461, 24851, 96921, 382358, 1522997, 6116518, 24740564, 100698617, 412126133, 1694982357, 7001729420, 29037602898, 120856092153, 504647152650, 2113469775619, 8875358529059, 37364827472930, 157668052571948, 666735804080597, 2825054673048981
Offset: 0

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Author

Paul D. Hanna, Jan 11 2023

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 127*x^5 + 460*x^6 + 1710*x^7 + 6461*x^8 + 24851*x^9 + 96921*x^10 + 382358*x^11 + 1522997*x^12 + ...
SPECIFIC VALUES.
A(2/9) = 2.24070435506724977359903344036738515875266644317987374...
A(x) = 2 at x = 0.21791735938682393028374635435485389216073583164032813...
A(1/5) = 1.63325728843716074555468074513852677972333543319428229...
A(1/6) = 1.36828627213340815002770404510072582545059876619425902...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
    A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (1 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #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) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k), for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 4.470597712126170109... and c = 1.18164918660560739... - Vaclav Kotesovec, Mar 14 2023