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A135080 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of x+x^2 (cf. A122888).

%I A135080 #11 Mar 30 2012 18:37:07
%S A135080 1,1,1,2,2,1,8,7,3,1,50,40,15,4,1,436,326,112,26,5,1,4912,3492,1128,
%T A135080 240,40,6,1,68098,46558,14373,2881,440,57,7,1,1122952,744320,221952,
%U A135080 42604,6135,728,77,8,1,21488640,13889080,4029915,748548,103326,11565,1120,100
%N A135080 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of x+x^2 (cf. A122888).
%H A135080 Paul D. Hanna, <a href="/A135080/b135080.txt">Table of n, a(n) for n=0..495 (rows 0..30)</a>
%F A135080  Columns may be generated by a method illustrated by triangles A187005, A187115, and A187120. The main diagonal of triangles A187005, A187115, and A187120, equals columns 0, 1, and 2, respectively.
%e A135080 Triangle begins:
%e A135080 1;
%e A135080 1, 1;
%e A135080 2, 2, 1;
%e A135080 8, 7, 3, 1;
%e A135080 50, 40, 15, 4, 1;
%e A135080 436, 326, 112, 26, 5, 1;
%e A135080 4912, 3492, 1128, 240, 40, 6, 1;
%e A135080 68098, 46558, 14373, 2881, 440, 57, 7, 1;
%e A135080 1122952, 744320, 221952, 42604, 6135, 728, 77, 8, 1;
%e A135080 21488640, 13889080, 4029915, 748548, 103326, 11565, 1120, 100, 9, 1; ...
%e A135080 Coefficients in iterations of (x+x^2) form table A122888:
%e A135080 1;
%e A135080 1, 1;
%e A135080 1, 2, 2, 1;
%e A135080 1, 3, 6, 9, 10, 8, 4, 1;
%e A135080 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
%e A135080 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
%e A135080 1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; ...
%e A135080 This triangle T transforms one diagonal in the above table into another;
%e A135080 start with the main diagonal of A122888, A112319, which begins:
%e A135080 [1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
%e A135080 then the transform T*A112319 equals A112317, which begins:
%e A135080 [1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028, ...];
%e A135080 and the transform T*A112317 equals A112320, which begins:
%e A135080 [1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].
%o A135080 (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+x^2+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]}
%o A135080 (PARI) /* Generate by method given in A187005, A187115, A187120 (faster): */
%o A135080 {T(n,k)=local(Ck=x);for(m=1,n-k+1,Ck=(1/x^k)*subst(truncate(x^k*Ck),x,x+x^2 +x*O(x^m)));polcoeff(Ck,n-k+1,x)}
%Y A135080 Cf. columns: A135081, A135082, A135083.
%Y A135080 Cf. related tables: A122888, A166900, A187005, A187115, A187120.
%Y A135080 Cf. related sequences: A112319, A112317, A112320, A187009.
%K A135080 nonn,tabl
%O A135080 0,4
%A A135080 _Paul D. Hanna_, Nov 18 2007
%E A135080 Added cross-reference; example corrected and name changed by _Paul D. Hanna_, Feb 04 2011