A132318 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.

1, 1, 1, 1, 2, 1, 1, 15, 15, 1, 1, 1024, 2046, 1024, 1, 1, 7048181, 60060682, 60060682, 7048181, 1, 1, 469389728563470, 72057594037927935, 143176408618728932, 72057594037927935, 469389728563470, 1, 1, 2954306864416502250656677496683

Offset: 0

Author

Paul D. Hanna, Aug 19 2007

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins:
1;
1,1;
1,2,1;
1,15,15,1;
1,1024,2046,1024,1;
1,7048181,60060682,60060682,7048181,1;
1,469389728563470,72057594037927935,143176408618728932,72057594037927935,469389728563470,1;
Examples:
T(2,1) = [x^(1*2)] (1+x)^2*(1+x^2) = 2;
T(3,1) = [x^(1*4)] (1+x)^4*(1+x^2)^2*(1+x^4) = 15;
T(4,3) = [x^(3*8)] (1+x)^8*(1+x^2)^4*(1+x^4)^2*(1+x^8) = 1024;
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.
		

Links

• Eric Weisstein, Mathworld, Series Multisection.

Crossrefs

Cf. A132317 (column 1), A132316.

Programs

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)))}

Formula

Row sums equal 2^(2^n - n) for n>0 - improved formula and proof by Max Alekseyev, Aug 19 2007.