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Search: id:A131791
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| A131791 |
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Triangle read by rows of 2^n terms for n>=0: let S(n) denote the initial 2^n terms of the partial sums of row n; starting with a single '1' in row 0, generate row n+1 by concatenating S(n) with the terms of S(n) when read in reverse order. |
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+0
2
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| 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 6, 5, 3, 1, 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1, 1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269, 270, 270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14, 5, 1, 1, 6, 20, 49, 99, 175, 280, 415
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row sums (and central terms) form A028361: Product_{i=0..n-1} (2^i + 1).
I'm interested in the graph of S(n). It appears to tend to a limit curve if scaled appropriately, e.g. scaled to fit a [0,1] box by f_n(x) = T(n,[x*2^n])/A028361(n-1). In this setup I think that the limit curve f(x) satisfies f(0)=0, f(1-x)=f(x), f(1/2)=1, f'(x)=2f(2x) for x<=1/2. Is this equation solvable? - Martin Fuller (martin_n_fuller(AT)btinternet.com), Aug 31 2007
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FORMULA
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T(n, 2^(n-1)) = A028361(n-1) for n>=1. T(n, 2^(n-2)) = A028362(n-1) for n>=2. Sum_{k=0..2^n-1} (k+1)*T(n,k) = A028362(n+1) for n>=0.
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 5, 6, 6, 5, 3, 1;
1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1; ...
1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269, 270, 270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14, 5, 1; ...
From row 2: [1,2,2,1], take the partial sums: [1,3,5,6] and concatenate to this the terms in reverse order: [6,5,3,1] to obtain row 3: [1,3,5,6, 6,5,3,1].
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PROGRAM
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(PARI) {T(n, k)=local(A=[1], B=[1]); if(n==0, 1, for(i=0, n-1, B=Vec(Ser(A)/(1-x)); A=concat(B, Vec(Pol(B)+O(x^#B)))); A[k+1])}
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CROSSREFS
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Cf. A131792 (main diagonal); A028361, A028362.
Sequence in context: A076037 A076263 A008302 this_sequence A010358 A010048 A055870
Adjacent sequences: A131788 A131789 A131790 this_sequence A131792 A131793 A131794
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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), Jul 15 2007
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