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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.

%I A132318 #6 Feb 16 2025 08:33:06
%S A132318 1,1,1,1,2,1,1,15,15,1,1,1024,2046,1024,1,1,7048181,60060682,60060682,
%T A132318 7048181,1,1,469389728563470,72057594037927935,143176408618728932,
%U A132318 72057594037927935,469389728563470,1,1,2954306864416502250656677496683
%N 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.
%C A132318 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 A132318 Eric Weisstein, Mathworld, <a href="https://mathworld.wolfram.com/SeriesMultisection.html">Series Multisection</a>.
%F A132318 Row sums equal 2^(2^n - n) for n>0 - improved formula and proof by _Max Alekseyev_, Aug 19 2007.
%e A132318 Triangle begins:
%e A132318 1;
%e A132318 1,1;
%e A132318 1,2,1;
%e A132318 1,15,15,1;
%e A132318 1,1024,2046,1024,1;
%e A132318 1,7048181,60060682,60060682,7048181,1;
%e A132318 1,469389728563470,72057594037927935,143176408618728932,72057594037927935,469389728563470,1;
%e A132318 Examples:
%e A132318 T(2,1) = [x^(1*2)] (1+x)^2*(1+x^2) = 2;
%e A132318 T(3,1) = [x^(1*4)] (1+x)^4*(1+x^2)^2*(1+x^4) = 15;
%e A132318 T(4,3) = [x^(3*8)] (1+x)^8*(1+x^2)^4*(1+x^4)^2*(1+x^8) = 1024;
%e A132318 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 A132318 (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 A132318 Cf. A132317 (column 1), A132316.
%K A132318 nonn,tabl
%O A132318 0,5
%A A132318 _Paul D. Hanna_, Aug 19 2007