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Search: id:A103525
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| A103525 |
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Triangle read by rows: T(n,k) is the coefficient of t^k (k>=0) in the polynomial P[n,t] defined by P[1,t]=P[2,t]=1, P[3,t]=1+t, P[n,t]=P[n-1,t]+P^2[n-2,1] for n>=4. |
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+0
1
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| 1, 1, 1, 1, 2, 1, 3, 3, 1, 7, 7, 2, 16, 25, 17, 6, 1, 65, 123, 94, 34, 5, 321, 923, 1263, 1076, 626, 254, 70, 12, 1, 4546, 16913, 28612, 28620, 18476, 7876, 2166, 352, 26, 107587, 609479, 1691387, 3050910, 4001833, 4044516, 3255042, 2126032, 1138124, 500806
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OFFSET
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1,5
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COMMENT
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T(n,k) is the number of certain types of trees (see the Duke et al. reference) of height n and having k branch nodes at level n-1. Row n has 2^(ceil(n/2)-2)+1 terms (n>=3). Row sums yield A000278. T(n,0)=A000278(n-1) for n>=2.
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REFERENCES
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W. Duke, Stephen J. Greenfield and Eugene R. Speer, Properties of a Quadratic Fibonacci Recurrence, J. Integer Sequences, 1998, #98.1.8.
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FORMULA
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T(1, 0)=1; T(2, 0)=1; T(3, 0)=T(3, 1)=1; T(n, k)=0 for k>=ceil(n/2); T(n, k)=T(n-1, k)+sum(T(n-2, j)T(n-2, k-j), j=0..k) for n>=4.
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EXAMPLE
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P[5,t]=3+3t+t^2; therefore T(3,0)=3, T(3,1)=3, T(3,2)=1.
Triangle begins:
1;
1;
1,1;
2,1;
3,3,1;
7,7,2;
16,25,17,6,1;
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MAPLE
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P[1]:=1:P[2]:=1:P[3]:=1+t:for n from 4 to 13 do P[n]:=sort(expand(P[n-1]+P[n-2]^2)) od:for n from 1 to 11 do seq(coeff(t*P[n], t^k), k=1..2^(ceil(n/2)-2)+1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000278.
Sequence in context: A066704 A165007 A127123 this_sequence A121436 A088074 A071463
Adjacent sequences: A103522 A103523 A103524 this_sequence A103526 A103527 A103528
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 21 2005
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