A380681 G.f. A(x) satisfies 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^n)^(2*n) with A(0) = 1.

1, 2, 7, 28, 122, 564, 2707, 13372, 67593, 347916, 1817244, 9607688, 51316353, 276485966, 1500906317, 8201237354, 45072131590, 248974971580, 1381605020061, 7698230495694, 43053083351989, 241588037857532, 1359801706168192, 7675245175662224, 43433685932208079, 246372179366760938, 1400586104620945564

Offset

0,2

Comments

Conjecture: if F(x) satisfies p*x = Sum_{n=-oo..+oo} (-1)^n * x^(p*n) * (F(x) + x^n)^(p*n) with F(0) = 1 for some fixed integer p, then F(x) is an integer series in x iff p is prime. The g.f. A(x) of this sequence is the case when p = 2; other cases include A380682 (p=3), A380683 (p=5), A380684 (p=7), A380685 (p=11), and A380686 (p=13).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 28*x^3 + 122*x^4 + 564*x^5 + 2707*x^6 + 13372*x^7 + 67593*x^8 + 347916*x^9 + 1817244*x^10 + ...
SPECIFIC VALUES.
A(t) = 8/3 at t = 0.16627953219622019805279570246940991078834981259924...
  where 2*t = Sum_{n=-oo..+oo} (-1)^n * t^(2*n) * (8/3 + t^n)^(2*n).
A(t) = 5/2 at t = 0.16593408096145587391708686610041478474400383166738...
A(t) = 7/3 at t = 0.16473188313622727999985924580149665711221079322425...
A(t) = 9/4 at t = 0.16370946812259108659437362613294469987579816905904...
A(t) = 2 at t = 0.1582554495689265312170108133077001693630299804203558...
  where 2*t = Sum_{n=-oo..+oo} (-1)^n * t^(2*n) * (2 + t^n)^(2*n).
A(t) = 9/5 at t = 0.15007119291270141115836901201779055345548840594439...
A(t) = 7/4 at t = 0.14725926290572441813998039990646564687997004739301...
A(t) = 5/3 at t = 0.14168681310432024176462735900085795243884220403741...
A(t) = 3/2 at t = 0.12621272596494869496681197467736861043686853250842...
  where 2*t = Sum_{n=-oo..+oo} (-1)^n * t^(2*n) * (3/2 + t^n)^(2*n).
A(1/7) = 1.6827477026530609643202045165072542778715322289260...
  where 2/7 = Sum_{n=-oo..+oo} (-1)^n * (1/7)^(2*n) * (A(1/7) + (1/7)^n)^(2*n).
A(1/8) = 1.4895459673576688630874970613372189105335022580657...
A(1/9) = 1.3868414638746031695292489403960233012774295034103...
A(1/10) = 1.321298156489101081144287622776587530373370762204...
A(1/12) = 1.241188717972373624507241574981876548049236351455...
  where 1/6 = Sum_{n=-oo..+oo} (-1)^n * (1/12)^(2*n) * (A(1/12) + (1/12)^n)^(2*n).

Prog

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

Crossrefs

Cf. A380682, A380683, A380684, A380685, A380686.

Sequence in context: A269228 A005435 A291091 * A215973 A143927 A253787

Adjacent sequences: A380678 A380679 A380680 * A380682 A380683 A380684

Keyword

nonn

Author

Paul D. Hanna, Jan 30 2025

Status

approved