A326605 G.f.: Sum_{n>=0} (2*n + 1) * x^n * (9 - x^n)^n.

1, 27, 402, 5103, 58959, 649539, 6907037, 71744535, 731768013, 7360989480, 73222111566, 721764371007, 7060733810570, 68630377364883, 663426925392564, 6382625095014309, 61149665581626645, 583701359488329915, 5553501498629257581, 52683216989246691471, 498464283739975769250

Offset

0,2

Comments

G.f. is congruent modulo 9 to Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)).

The cube root of the g.f. A(x) is an integer series (cf. A326606), and is congruent modulo 3 to Product_{n>=1} 1 - x^(2*n).

Links

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

Formula

G.f.: Sum_{n>=0} (2*n+1) * x^n * (9 - x^n)^n.

G.f.: Sum_{n>=0} (-1)^n * (2*n+1 + 9*x^(n+1)) * x^(n*(n+1)) / (1 - 9*x^(n+1))^(n+2).

a(n) ~ 2*n*3^(2*n). - Vaclav Kotesovec, Oct 09 2019

Example

G.f.: A(x) = 1 + 27*x + 402*x^2 + 5103*x^3 + 58959*x^4 + 649539*x^5 + 6907037*x^6 + 71744535*x^7 + 731768013*x^8 + 7360989480*x^9 + 73222111566*x^10 + ...
such that
A(x) = 1 + 3*x*(9-x) + 5*x^2*(9-x^2)^2 + 7*x^3*(9-x^3)^3 + 9*x^4*(9-x^4)^4 + 11*x^5*(9-x^5)^5 + 13*x^6*(9-x^6)^6 + 15*x^7*(9-x^7)^7 + ...
Also,
A(x) = (1 + 9*x)/(1 - 9*x)^2 - (3 + 9*x^2)*x^2/(1 - 9*x^2)^3 + (5 + 9*x^3)*x^6/(1 - 9*x^3)^4 - (7 + 9*x^4)*x^12/(1 - 9*x^4)^5 + (9 + 9*x^5)*x^20/(1 - 9*x^5)^6 - (11 + 9*x^6)*x^30/(1 - 9*x^6)^7 + (13 + 9*x^7)*x^42/(1 - 9*x^7)^8 + ...
RELATED SERIES.
A(x) is congruent modulo 9 to
A(x) (mod 9) = 1 - 3*x^2 + 5*x^6 - 7*x^12 + 9*x^20 - 11*x^30 + 13*x^42 - 15*x^56 + 17*x^72 - 19*x^90 + 21*x^110 + ... + (-1)^n*(2*n+1)*x^(n*(n+1)) + ...
The cube root of the g.f. is an integer series:
A(x)^(1/3) = 1 + 9*x + 53*x^2 + 504*x^3 + 3479*x^4 + 34362*x^5 + 248799*x^6 + 2483091*x^7 + 18383088*x^8 + 185472450*x^9 + 1378756330*x^10 + ... + A326606(n)*x^n + ...
which is congruent modulo 3 to Product_{n>=1} 1 - x^(2*n),
A(x)^(1/3) (mod 3) = 1 - x^2 - x^4 + x^10 + x^14 - x^24 - x^30 + x^44 + x^52 - x^70 - x^80 + x^102 + x^114 - x^140 - x^154 + x^184 + x^200 + ...

Prog

(PARI) /* By definition */
{a(n) = my(A = sum(m=0, n, (2*m + 1) * x^m * (9 - x^m + x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Accelerated series */
{a(n) = my(A = sum(m=0, sqrtint(n+1), (-1)^m * (2*m + 1 + 9*x^(m+1))* x^(m*(m+1)) / (1 - 9*x^(m+1) + x*O(x^n))^(m+2) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A326606.

Sequence in context: A251770 A033280 A125462 * A296853 A036222 A022655

Adjacent sequences: A326602 A326603 A326604 * A326606 A326607 A326608

Keyword

nonn

Author

Paul D. Hanna, Oct 08 2019

Status

approved