A132318 - OEIS

%I #5 Mar 30 2012 18:37:04

%S 1,1,1,1,2,1,1,15,15,1,1,1024,2046,1024,1,1,7048181,60060682,60060682,

%T 7048181,1,1,469389728563470,72057594037927935,143176408618728932,

%U 72057594037927935,469389728563470,1,1,2954306864416502250656677496683

%N Triangle, read by rows, where T(n,k) = [x^(k*2^(n-1))] Product_{i=0..n-1} (1 + x^(2^i))^(2^(n-i-1)) for n>0 with T(0,0)=1.

%C There are n*2^(n-1)+1 coefficients in P(n) = Product_{i=0..n-1} (1 + x^(2^i))^(2^(n-i-1)) for n>0; in this triangle, row n consists of coefficients of x^(k*2^(n-1)) in P(n) as k=0..n.

%H Eric Weisstein, Mathworld, <a href="http://mathworld.wolfram.com/SeriesMultisection.html">Series Multisection</a>.

%F Row sums equal 2^(2^n - n) for n>0 - improved formula and proof by _Max Alekseyev_, Aug 19 2007.

%e Triangle begins:

%e 1;

%e 1,1;

%e 1,2,1;

%e 1,15,15,1;

%e 1,1024,2046,1024,1;

%e 1,7048181,60060682,60060682,7048181,1;

%e 1,469389728563470,72057594037927935,143176408618728932,72057594037927935,469389728563470,1;

%e Examples:

%e T(2,1) = [x^(1*2)] (1+x)^2*(1+x^2) = 2;

%e T(3,1) = [x^(1*4)] (1+x)^4*(1+x^2)^2*(1+x^4) = 15;

%e T(4,3) = [x^(3*8)] (1+x)^8*(1+x^2)^4*(1+x^4)^2*(1+x^8) = 1024;

%e T(5,3) = [x^(3*16)] (1+x)^16*(1+x^2)^8*(1+x^4)^4*(1+x^8)^2*(1+x^16) = 60060682.

%o (PARI) {T(n,k)=if(n==0,1,polcoeff(prod(i=0,n-1,(1+x^(2^i)+x*O(x^(k*2^(n-1))))^(2^(n-i-1))),k*2^(n-1)))}

%Y Cf. A132317 (column 1), A132316.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Aug 19 2007