A350411 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: 0 = [x^n] Sum_{m=0..n+1} A(x)^(m^2) / m!, for n > 1.

-1, 1, 6, 76, 74, 26058, -7794632, -14388774920, 213380711849598, 28487519401892735386, -38029127603565193657832846, -558442679082054075230555562505128, 98405860538000983882824515617294391388622, 225427456282611271896161167867656072505435638401710

Offset

0,3

Comments

This sequence is conjectured to consist entirely of integers.

Links

Table of n, a(n) for n=0..13.

Example

G.f.: A(x) = -1 + x + 6*x^2 + 76*x^3 + 74*x^4 + 26058*x^5 - 7794632*x^6 - 14388774920*x^7 + 213380711849598*x^8 + 28487519401892735386*x^9 + ...
The table of coefficients of x^k in A(x)^(n^2), k >= 0, begins:
n=1: [-1, 1, 6, 76, 74, 26058, -7794632, ...];
n=2: [1, -4, -18, -236, 761, -99192, 31519032, ...];
n=3: [-1, 9, 18, 336, -4716, 221688, -72062976, ...];
n=4: [1, -16, 24, -336, 13116, -438576, 130813640, ...];
n=5: [-1, 25, -150, 600, -25800, 855780, -210360800, ...];
n=6: [1, -36, 414, -2316, 46161, -1632600, 316244424, ...];
n=7: [-1, 49, -882, 8036, -97706, 3042606, -459537288, ...]; ...
in which, by definition, the following sums along the columns equal zero:
0 = (6)/1! + (-18)/2! + (18)/3! ;
0 = (76)/1! + (-236)/2! + (336)/3! + (-336)/4! ;
0 = (74)/1! + (761)/2! + (-4716)/3! + (13116)/4! + (-25800)/5! ;
0 = (26058)/1! + (-99192)/2! + (221688)/3! + (-438576)/4! + (855780)/5! + (-1632600)/6! ;
0 = (-7794632)/1! + (31519032)/2! + (-72062976)/3! + (130813640)/4! + (-210360800)/5! + (316244424)/6! + (-459537288)/7! ; ...
One may continue the above pattern to determine all the terms of this sequence.

Prog

(PARI) {a(n) = my(A=[-1, 1, 0]); for(i=1, n, A=concat(A, 0);
A[#A-1] = (-1)^(#A-1) * (#A-2)! * polcoeff( sum(m=0, #A-1, Ser(A)^(m^2) / m!) , #A-2) ); A[n+1]}
for(n=0, 16, print1(a(n), ", "))

Crossrefs

Cf. A350410, A350412.

Sequence in context: A336228 A326011 A350283 * A263228 A229571 A016090

Adjacent sequences: A350408 A350409 A350410 * A350412 A350413 A350414

Keyword

sign

Author

Paul D. Hanna, Dec 29 2021

Status

approved