0,3

Amazingly, this sequence also equals column 0 of tables A135878 and A135879, which have unusual recurrences seemingly unrelated to triangle A135880.

Paul D. Hanna, Table of n, a(n) for n = 0..100

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.

(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]}

Cf. A135880, A135879, A135878; other columns: A135882, A135883, A135884.

Sequence in context: A020100 A128230 A084784 this_sequence A007815 A109286 A009466

Adjacent sequences: A135878 A135879 A135880 this_sequence A135882 A135883 A135884

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 15 2007

Typo in entries (false comma) corrected by N. J. A. Sloane (njas(AT)research.att.com), Jan 23 2008