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%I #5 Mar 30 2012 18:37:07
%S 1,1,2,6,25,138,970,8390,86796,1049546,14563135,228448504,4002300038,
%T 77523038603,1646131568618,38043008887356,950967024783228,
%U 25573831547118764,736404945614783668,22611026430036582671
%N Column 0 of triangle A135880.
%C Amazingly, this sequence also equals column 0 of tables A135878 and A135879, which have unusual recurrences seemingly unrelated to triangle A135880.
%H Paul D. Hanna, <a href="/A135881/b135881.txt">Table of n, a(n) for n = 0..100</a>
%e Equals column 0 of triangle P=A135880:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 6, 7, 3, 1;
%e 25, 34, 15, 4, 1;
%e 138, 215, 99, 26, 5, 1;
%e 970, 1698, 814, 216, 40, 6, 1;
%e 8390, 16220, 8057, 2171, 400, 57, 7, 1; ...
%e where column k of P^2 equals column 0 of P^(2k+2)
%e such that column 0 of P^2 equals this sequence shift left.
%e Also equals column 0 of irregular triangle A135879:
%e 1;
%e 1,1;
%e 2,2,1,1;
%e 6,6,4,4,2,2,1;
%e 25,25,19,19,13,13,9,5,5,3,1,1;
%e 138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1; ...
%e which has a recurrence similar to that of triangle A135877
%e which generates the double factorials.
%o (PARI) /* Generated as column 0 in triangle A135880: */ {a(n)=local(P=Mat(1),R,PShR);if(n==0,1,for(i=0,n, PShR=matrix(#P,#P, r,c, if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c],if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1]))));P[n+1,1])}
%o (PARI) /* Generated as column 0 in triangle A135879 (faster): */ {a(n)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[]; for(j=1,#A,if(j+m-1==floor((m+2)^2/4)-1,m+=1;B=concat(B,0));B=concat(B,A[ j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));A[1]}
%Y Cf. A135880, A135879, A135878; other columns: A135882, A135883, A135884.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 15 2007
%E Typo in entries (false comma) corrected by _N. J. A. Sloane_, Jan 23 2008
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