A136219 Number of terms in rows of irregular triangle A136218.

1, 3, 7, 13, 22, 33, 47, 64, 84, 106, 131, 159, 190, 224, 261, 301, 343, 388, 436, 487, 541, 598, 658, 721, 787, 856, 928, 1003, 1081, 1162, 1245, 1331, 1420, 1512, 1607, 1705, 1806, 1910, 2017, 2127, 2240, 2356, 2475, 2597, 2722, 2850, 2981, 3115, 3252

Offset

0,2

Comments

A136218 is a triangle where row n+1 is generated from row n by first inserting zeros in row n at positions {[m*(m+7)/6], m>=0} and then taking partial sums, starting with a '1' in row 0.

Links

Table of n, a(n) for n=0..48.

Formula

G.f. A(x) = (1+x+x^2)/(1-x)^3 - [Sum_{n>=0} x^b(n)]/(1-x)^2 where exponents b(n) = A136169(n) satisfy: b(n) = 2*b(n-1) - [(n+1)/3] for n>0 with b(0)=1 and the g.f. of the exponents is B(z) = [1 - z^2*(1+z+z^2)/(1-z^3)^2]/(1-2*z).

Example

G.f. A(x) = (1+x+x^2)/(1-x)^3 - (x+x^2+x^3+x^5+x^9+x^16+x^30+x^58+...)/(1-x)^2.

Prog

(PARI) {a(n)=local(A, X=x+x*O(x^n), bd=#binary(2*n), B=(1 - x^2*(1+x+x^2)/(1-x^3+x*O(x^bd))^2 )/(1-2*x)); A=(1+x+x^2)/(1-X)^3 - sum(k=0, bd, x^polcoeff(B, k))/(1-X)^2; polcoeff(A, n)}

Crossrefs

Cf. A136218; A136169.

Sequence in context: A147464 A146058 A155354 * A078582 A051336 A253896

Adjacent sequences: A136216 A136217 A136218 * A136220 A136221 A136222

Keyword

nonn

Author

Paul D. Hanna, Dec 23 2007

Status

approved