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%I #28 Sep 22 2020 00:25:55
%S 1,0,2,0,2,4,0,0,8,8,0,0,4,24,16,0,0,0,24,64,32,0,0,0,8,96,160,64,0,0,
%T 0,0,64,320,384,128,0,0,0,0,16,320,960,896,256,0,0,0,0,0,160,1280,
%U 2688,2048,512,0,0,0,0,0,32,960,4480,7168,4608,1024,0,0,0,0,0,0,384,4480,14336,18432,10240,2048
%N Riordan array (1, 2+2x).
%C Row sums give A002605. Diagonal sums give A052907.
%C The Riordan array (1,s+t*x) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
%C T(n,k) is the number of compositions of n into two types of parts of size 1 and 2 that have exactly k parts. - _Geoffrey Critzer_, Aug 18 2012.
%C Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Sep 22 2020
%H G. C. Greubel, <a href="/A099040/b099040.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F Number triangle T(n, k) = 2^k*binomial(k, n-k).
%F Columns have g.f. (2x+2x^2)^k.
%F T(n,k) = A026729(n,k)*2^k. - _Philippe Deléham_, Jul 28 2006
%F O.g.f.: 1/(1-2*y*x-2*y*x^2). - _Geoffrey Critzer_, Aug 18 2012.
%e Rows begin {1}, {0,2}, {0,2,4}, {0,0,8,8}, {0,0,4,24,16}, {0,0,0,24,64,32},...
%e T(3,2)=8 because we have: 1+2,1+2',1'+2,1'+2',2+1,2+1',2'+1,2'+1' where a part of the second type is designated by '. - _Geoffrey Critzer_, Aug 18 2012
%t nn = 8; CoefficientList[Series[1/(1 - 2 y x - 2 y x^2), {x, 0, nn}], {x, y}] // Grid (* _Geoffrey Critzer_, Aug 18 2012 *)
%Y Cf. A002605, A026729, A038208, A052907.
%K easy,nonn,tabl
%O 0,3
%A _Paul Barry_, Sep 23 2004
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