A381362 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(2*n-1) * (x^n + A(x))^(2*n-1).

1, 2, 46, 862, 20414, 526106, 14519710, 419293310, 12527971550, 384222183226, 12030729376882, 383113013296770, 12372095284443242, 404291094649795558, 13345757405802263098, 444433438912442427974, 14914705697211799893458, 503945427634033914776682, 17131542722554038753304418

Offset

0,2

Comments

Conjecture: for n > 0, a(n) == 2 (mod 4).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..260

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) 1/2 = Sum_{n=-oo..+oo} (x*A(x))^n * (A(x)^n + x)^(2*n-1) * (x^n + A(x))^(2*n-1).

(2) 1/2 = Sum_{n=-oo..+oo} (x*A(x))^(2*n^2) / ( (1 + x*A(x)^n)^(2*n+1) * (1 + x^n*A(x))^(2*n+1) ).

Example

G.f.: A(x) = 1 + 2*x + 46*x^2 + 862*x^3 + 20414*x^4 + 526106*x^5 + 14519710*x^6 + 419293310*x^7 + 12527971550*x^8 + 384222183226*x^9 + ...

Prog

(PARI) {a(n) = my(V=[1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( 4*sum(m=-#V, #V, x^m*A^m * (A^m + x)^(2*m-1) * (A + x^m)^(2*m-1) ), #V-1); ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A380066, A381363, A381364, A381365.

Sequence in context: A302949 A174592 A066555 * A290046 A012001 A012006

Adjacent sequences: A381349 A381350 A381351 * A381363 A381364 A381365

Keyword

nonn,new

Author

Paul D. Hanna, Feb 21 2025

Status

approved