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 A117401 Triangle, read by rows, defined by: T(n,k) = 2^((n-k)*k) for n>=k>=0. 13
 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; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. LINKS Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. FORMULA 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, Apr 21 2008 T(n,k)=1/n*[2^(n-k)*k*T(n-1,k-1) + 2^k*(n-k)*T(n-1,k)], where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014 EXAMPLE 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; ... MATHEMATICA Table[2^((n-k)k), {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Jan 09 2017 *) PROG (PARI) T(n, k)=if(n

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Last modified October 23 22:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)