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 A135881 Column 0 of triangle A135880. 12
 1, 1, 2, 6, 25, 138, 970, 8390, 86796, 1049546, 14563135, 228448504, 4002300038, 77523038603, 1646131568618, 38043008887356, 950967024783228, 25573831547118764, 736404945614783668, 22611026430036582671 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Amazingly, this sequence also equals column 0 of tables A135878 and A135879, which have unusual recurrences seemingly unrelated to triangle A135880. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..100 EXAMPLE 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. PROG (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]} CROSSREFS Cf. A135880, A135879, A135878; other columns: A135882, A135883, A135884. Sequence in context: A084784 A255841 A197772 * A007815 A195259 A292748 Adjacent sequences:  A135878 A135879 A135880 * A135882 A135883 A135884 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 15 2007 EXTENSIONS Typo in entries (false comma) corrected by N. J. A. Sloane, Jan 23 2008 STATUS approved

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Last modified August 9 04:18 EDT 2022. Contains 356016 sequences. (Running on oeis4.)