A326003 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.

1, 3, 9, 10, 27, 21, 64, 36, 105, 85, 171, 78, 359, 105, 372, 346, 573, 171, 1105, 210, 1116, 1009, 1134, 300, 3237, 456, 1743, 2386, 3375, 465, 5947, 528, 5529, 4885, 3537, 1926, 14917, 741, 4770, 9010, 16551, 903, 17963, 990, 19977, 22291, 8028, 1176, 49925, 1527, 23961, 24634, 41289, 1485, 48502, 27336, 58809, 37621, 15255, 1830, 184218, 1953, 18384, 59830, 106137, 77286, 121705, 2346, 140115, 78385, 159846, 2628, 346846, 2775, 30267, 293866

Offset

0,2

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),

for any fixed integer k; here, k = 3 and p = 1, q = x, r = x.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..2050

Formula

Generating functions.

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

(2) Sum_{n>=0} (n+1)*(n+2)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+3).

Example

G.f.: A(x) = 1 + 3*x + 9*x^2 + 10*x^3 + 27*x^4 + 21*x^5 + 64*x^6 + 36*x^7 + 105*x^8 + 85*x^9 + 171*x^10 + 78*x^11 + 359*x^12 + 105*x^13 + 372*x^14 + 346*x^15 + 573*x^16 + 171*x^17 + 1105*x^18 + 210*x^19 + 1116*x^20 + ...
where we have the following series identity:
A(x) = 1 + 3*x*(1+x) + 6*x^2*(1+x^2)^2 + 10*x^3*(1+x^3)^3 + 15*x^4*(1+x^4)^4 + 21*x^5*(1+x^5)^5  + 28*x^6*(1+x^6)^6 + 36*x^7*(1+x^7)^7 + 45*x^8*(1+x^8)^8 + 55*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^3 + 3*x^2/(1-x^2)^4 + 6*x^6/(1-x^3)^5 + 10*x^12/(1-x^4)^6 + 15*x^20/(1-x^5)^7 + 21*x^30/(1-x^6)^8 + 28*x^42/(1-x^7)^9 + 36*x^56/(1-x^8)^10 +...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)/2 * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)/2 * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3))); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))

Crossrefs

Cf. A217668 (k=1), A326002 (k=2), A326004 (k=4), A326005 (k=5).

Sequence in context: A247519 A344295 A077560 * A060140 A340387 A025616

Adjacent sequences: A326000 A326001 A326002 * A326004 A326005 A326006

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2019

Status

approved