A325286 G.f. satisfies: 1 = Sum_{n>=0} (1+x)^(n*(n-1)/2) / A(x)^n * 1/2^(n+1).

1, 1, 3, 25, 343, 6441, 150975, 4203201, 134852079, 4886641681, 197154406591, 8760602600193, 425074860993439, 22363792326962881, 1268239233311498079, 77129745316500047745, 5008173999379887257151, 345838251972031108425345, 25309861534968595801377279, 1956926079593452273940279169, 159406563966400881627947865279, 13645204581985719926987977747329, 1224591755319676016226530026499583, 114980206425267526899287638805977857

Offset

0,3

Comments

a(n) == 1 (mod 2) for n >= 0.

Conjecture: a(n) == 1 (mod 3) iff n is a number whose base-3 representation contains no 2 (cf. A005836), otherwise a(n) == 0 (mod 3).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..175

Formula

G.f. satisfies:

(1) 1 = Sum_{n>=0} (1+x)^(n*(n-1)/2) / A(x)^n * 1/2^(n+1).

(2) 1 = Sum_{n>=0} (1+x)^(n*(n+1)/2) / A(x)^(n+1) * 1/2^(n+1).

Example

G.f.: A(x) = 1 + x + 3*x^2 + 25*x^3 + 343*x^4 + 6441*x^5 + 150975*x^6 + 4203201*x^7 + 134852079*x^8 + 4886641681*x^9 + 197154406591*x^10 + ...
such that
1 = 1/2 + 1/(2^2*A(x)) + (1+x)/(2^3*A(x)^2) + (1+x)^3/(2^4*A(x)^3) + (1+x)^6/(2^5*A(x)^4) + (1+x)^10/(2^6*A(x)^5) + (1+x)^15/(2^7*A(x)^6) + (1+x)^21/(2^8*A(x)^7) + ...
also,
1 = 1/(2*A(x)) + (1+x)/(2*A(x))^2 + (1+x)^3/(2*A(x))^3 + (1+x)^6/(2*A(x))^4 + (1+x)^10/(2*A(x))^5 + (1+x)^15/(2*A(x))^6 + (1+x)^21/(2*A(x))^7 + (1+x)^28/(2*A(x))^8 + ...

Prog

(PARI) /* Requires adequate precision */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = round( polcoeff( sum(m=0, 10*#A+100, (1+x+x*O(x^#A))^(m*(m-1)/2)/Ser(A)^m/2^(m+1)*1.), #A-1))); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A303290.

Sequence in context: A023997 A154961 A322760 * A085527 A093360 A161629

Adjacent sequences: A325283 A325284 A325285 * A325287 A325288 A325289

Keyword

nonn

Author

Paul D. Hanna, Apr 18 2019

Status

approved