Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #4 Sep 13 2013 19:04:35
%S 1,2,1,9,4,1,78,30,6,1,1038,364,63,8,1,18968,6233,986,108,10,1,443595,
%T 139008,20685,2072,165,12,1,12681960,3833052,545736,51494,3750,234,14,
%U 1,429244197,126105168,17365336,1569920,107760,6148,315,16,1,16801151910
%N Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x*(1+x)^2 (cf. A166888).
%e Triangle begins:
%e 1;
%e 2,1;
%e 9,4,1;
%e 78,30,6,1;
%e 1038,364,63,8,1;
%e 18968,6233,986,108,10,1;
%e 443595,139008,20685,2072,165,12,1;
%e 12681960,3833052,545736,51494,3750,234,14,1;
%e 429244197,126105168,17365336,1569920,107760,6148,315,16,1;
%e 16801151910,4824243516,647216568,56661004,3728952,200583,9394,408,18,1;
%e 746998729887,210489178476,27653205177,2361036896,150566205,7768320,343063,13616,513,20,1;
%e 37200237947376,10318212622770,1332422277828,111501524409,6938694600,347030328,14703080,550300,18942,630,22,1; ...
%e Coefficients in iterations of x*(1+x)^2 form table A166888:
%e 1;
%e 1,2,1;
%e 1,4,10,18,23,22,15,6,1;
%e 1,6,27,102,333,960,2472,5748,12150,23388,40926,64872,92772,...;
%e 1,8,52,300,1578,7692,35094,150978,615939,2393628,8892054,...;
%e 1,10,85,660,4790,32920,215988,1360638,8265613,48585702,...;
%e 1,12,126,1230,11385,101010,864813,7178700,57976074,456783888,...;
%e 1,14,175,2058,23163,251832,2660028,27405798,276215313,...;
%e 1,16,232,3192,42308,544600,6842220,84191772,1017153322,...;
%e ...
%e This triangle T transforms one diagonal in A166888 into another,
%e for example: T * A154256 = A119820, T * A119820 = A166889, where
%e A154256 = [1,2,10,102,1578,32920,864813,27405798,1017153322,...];
%e A119820 = [1,4,27,300,4790,101010,2660028,84191772,3115739358,...];
%e A166889 = [1,6,52,660,11385,251832,6842220,221228244,8311401351,...].
%o (PARI) {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%Y Cf. columns: A166891, A166892, A166893; A229113 (row sums).
%Y Cf. variants: A135080, A166884.
%Y Cf. A166888, A154256, A119820, A166889.
%K nonn,tabl
%O 1,2
%A _Paul D. Hanna_, Nov 22 2009