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Search: id:A117401
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| A117401 |
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Triangle, read by rows, defined by: T(n,k) = 2^((n-k)*k) for n>=k>=0. |
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+0
12
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| 1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 8, 1, 1, 16, 64, 64, 16, 1, 1, 32, 256, 512, 256, 32, 1, 1, 64, 1024, 4096, 4096, 1024, 64, 1, 1, 128, 4096, 32768, 65536, 32768, 4096, 128, 1, 1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1, 1, 512, 65536
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character.
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REFERENCES
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Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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FORMULA
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G.f.: A(x,y) = Sum_{n>=0} x^n/(1-2^n*x*y). G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,2*y).
Equals ConvOffsStoT transform of the 2^n series: (1, 2, 4, 8,...); e.g., ConvOffs transform of (1, 2, 4, 8) = (1, 8, 16, 8, 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 21 2008
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EXAMPLE
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A(x,y) = 1/(1-xy) + x/(1-2xy) + x^2/(1-4xy) + x^3/(1-8xy) + ...
Triangle begins:
1;
1,1;
1,2,1;
1,4,4,1;
1,8,16,8,1;
1,16,64,64,16,1;
1,32,256,512,256,32,1;
1,64,1024,4096,4096,1024,64,1;
1,128,4096,32768,65536,32768,4096,128,1;
1,256,16384,262144,1048576,1048576,262144,16384,256,1; ...
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PROGRAM
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(PARI) T(n, k)=if(n<k|k<0, 0, 2^((n-k)*k))
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CROSSREFS
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Cf. A117402 (row sums), A117403 (antidiagonal sums), A002416 (central terms).
Sequence in context: A154867 A064298 A099594 this_sequence A144324 A034372 A155971
Adjacent sequences: A117398 A117399 A117400 this_sequence A117402 A117403 A117404
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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), Mar 12 2006
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