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A135881
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Column 0 of triangle A135880.
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12
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1, 1, 2, 6, 25, 138, 970, 8390, 86796, 1049546, 14563135, 228448504, 4002300038, 77523038603, 1646131568618, 38043008887356, 950967024783228, 25573831547118764, 736404945614783668, 22611026430036582671
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OFFSET
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0,3
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COMMENTS
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Amazingly, this sequence also equals column 0 of tables A135878 and A135879, which have unusual recurrences seemingly unrelated to triangle A135880.
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LINKS
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EXAMPLE
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Equals column 0 of triangle P=A135880:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1;
8390, 16220, 8057, 2171, 400, 57, 7, 1; ...
where column k of P^2 equals column 0 of P^(2k+2)
such that column 0 of P^2 equals this sequence shift left.
Also equals column 0 of irregular triangle A135879:
1;
1,1;
2,2,1,1;
6,6,4,4,2,2,1;
25,25,19,19,13,13,9,5,5,3,1,1;
138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1; ...
which has a recurrence similar to that of triangle A135877
which generates the double factorials.
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PROG
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(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])}
(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]}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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