A136219 - OEIS

%I #3 Mar 30 2012 18:37:08

%S 1,3,7,13,22,33,47,64,84,106,131,159,190,224,261,301,343,388,436,487,

%T 541,598,658,721,787,856,928,1003,1081,1162,1245,1331,1420,1512,1607,

%U 1705,1806,1910,2017,2127,2240,2356,2475,2597,2722,2850,2981,3115,3252

%N Number of terms in rows of irregular triangle A136218.

%C 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.

%F 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).

%e 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.

%o (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)}

%Y Cf. A136218; A136169.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 23 2007