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Search: id:A132318
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| A132318 |
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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. |
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3
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| 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
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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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.
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LINKS
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Eric Weisstein, Mathworld, Series Multisection.
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FORMULA
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Row sums equal 2^(2^n - n) for n>0 - improved formula and proof by Max Alekseyev (maxal(AT)cs.ucsd.edu), Aug 19 2007.
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EXAMPLE
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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.
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PROGRAM
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(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)))}
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CROSSREFS
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Cf. A132317 (column 1), A132316.
Sequence in context: A054505 A132610 A132625 this_sequence A078089 A095836 A090163
Adjacent sequences: A132315 A132316 A132317 this_sequence A132319 A132320 A132321
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 19 2007
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