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%I #5 Mar 30 2012 18:37:42
%S 1,0,1,1,0,2,0,2,0,1,3,0,10,0,2,0,23,0,20,0,2,45,0,196,0,70,0,4,0,132,
%T 0,154,0,28,0,1,315,0,1636,0,798,0,84,0,2,0,5067,0,7180,0,1806,0,120,
%U 0,2,14175,0,83754,0,50270,0,7392,0,330,0,4,0,146430,0,239327,0,74800,0
%N Triangle of coefficients T[n,m] of polynomials n, n^2, (n+2n^3)/3, n^2(2+n^2)/3, n(3+10n^2+2n^4)/15, etc. after multiplication by the denominators (A049606).
%C These polynomials are P(1, n) = 2*Sum[k, {k,1,n-1}] + n, counting up to n and down again; P(2, m) = 2*Sum[P(1,n), {n,1,m-1}] + P(1,m), meaning up and down to n and this for n from 1 up to m and down again; etc.
%e 1+2+3+2+1 = 3^2, (1)+(1+2+1)+(1+2+3+2+1)+(1+2+1)+(1) = (n+2n^3)/3.
%t CoefficientList[ #, n ]&/@(NestList[ ((2*Sum[ #, {n, k-1} ]+(#/. n->k)//Simplify)/.k->n)&, n, -1+16 ] Denominator[ 2^#/#!&/@Range[ 16 ] ])
%Y Row sums give A049606 again, final entry in each row seems to give A048896.
%K nonn,tabl
%O 1,6
%A _Wouter Meeussen_, Oct 30 2001
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