A326269 G.f.: Sum_{n>=0} (1+x + x^n)^n * x^n.

1, 1, 3, 3, 7, 10, 17, 27, 41, 70, 109, 168, 276, 439, 688, 1099, 1774, 2820, 4488, 7219, 11596, 18574, 29844, 48040, 77302, 124515, 200756, 323695, 522168, 843020, 1361409, 2198679, 3552094, 5740668, 9279009, 14999925, 24252057, 39216310, 63419775, 102569373, 165898349, 268344639, 434076911, 702197193, 1135967897, 1837747824, 2973155053, 4810149922, 7782281092, 12591037633, 20371441356

Offset

0,3

Comments

More generally, the following sums are equal:

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

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

for any fixed integer k; this sequence results when k=1, p = 1+x, q = x, r = x.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

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

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

a(n) ~ (5 + sqrt(5))/10 * Phi^n, where Phi = (1 + sqrt(5))/2.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 27*x^7 + 41*x^8 + 70*x^9 + 109*x^10 + 168*x^11 + 276*x^12 + 439*x^13 + 688*x^14 + 1099*x^15 + ...
such that
A(x) = 1 + (1+2*x)*x + (1+x+x^2)^2*x^2 + (1+x+x^3)^3*x^3 + (1+x+x^4)^4*x^4 + (1+x+x^5)^5*x^5 + (1+x+x^6)^6*x^6 + (1+x+x^7)^7*x^7 + (1+x+x^8)^8*x^8 + ...
also
A(x) = 1/(1-x-x^2) + x^2/(1-x^2-x^3)^2 + x^6/(1-x^3-x^4)^3 + x^12/(1-x^4-x^5)^4 + x^20/(1-x^5-x^6)^5 + x^30/(1-x^6-x^7)^6 + x^42/(1-x^7-x^8)^7 + ...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, (1+x + x^m +x*O(x^n))^m * x^m ) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, sqrtint(n+1), x^(m*(m+1)) / (1 - x^(m+1) - x^(m+2) +x*O(x^n) )^(m+1) ) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

Crossrefs

Sequence in context: A350394 A013915 A136445 * A052989 A358823 A252750

Adjacent sequences: A326266 A326267 A326268 * A326270 A326271 A326272

Keyword

nonn

Author

Paul D. Hanna, Jul 02 2019

Status

approved