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%I #11 Aug 10 2015 23:51:03
%S 1,1,2,1,4,7,1,6,20,40,1,8,39,138,326,1,10,64,318,1258,3492,1,12,95,
%T 604,3242,14476,46558,1,14,132,1020,6844,40348,202655,744320,1,16,175,
%U 1590,12750,92140,598083,3354848,13889080,1,18,224,2338,21766,185240,1450388,10337402,64246776,296459376
%N Triangle, read by rows, where row n equals the coefficients of y^k in R_{n-1}(y+y^2) for k=2..n, where R_n(y) is the n-th row polynomial in y for n>=2 with R_2(y)=y^2.
%F T(n,k) = Sum_{j=[k/2],k} C(j,k-j)*T(n-1,j) for n>=2, k=2..n, with T(n,2)=1 and T(n,k)=0 when k>n or k<2.
%F Main diagonal equals column 1 of triangle A135080, which transforms diagonals in the table of coefficients of the iterations of x+x^2.
%F Triangle A135080 also transforms diagonals in this triangle into each other.
%F Diagonal m of this triangle equals column 1 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 4, 7;
%e 1, 6, 20, 40;
%e 1, 8, 39, 138, 326;
%e 1, 10, 64, 318, 1258, 3492;
%e 1, 12, 95, 604, 3242, 14476, 46558;
%e 1, 14, 132, 1020, 6844, 40348, 202655, 744320;
%e 1, 16, 175, 1590, 12750, 92140, 598083, 3354848, 13889080;
%e 1, 18, 224, 2338, 21766, 185240, 1450388, 10337402, 64246776, 296459376;
%e 1, 20, 279, 3288, 34818, 340112, 3097162, 26277556, 204706486, 1398909332, 7125938790; ...
%e in which rows can be generated as illustrated below.
%e Row polynomials R_n(y), n>=2, begin:
%e R_2(y) = y^2;
%e R_3(y) = y^2 + 2*y^3;
%e R_4(y) = y^2 + 4*y^3 + 7*y^4;
%e R_5(y) = y^2 + 6*y^3 + 20*y^4 + 40*y^5;
%e R_6(y) = y^2 + 8*y^3 + 39*y^4 + 138*y^5 + 326*y^6; ...
%e where row n = coefficients of y^k in R_{n-1}(y+y^2) for k=2..n;
%e this method is illustrated by:
%e n=3: R_2(y+y^2) = (y^2 + 2*y^3) + y^4;
%e n=4: R_3(y+y^2) = (y^2 + 4*y^3 + 7*y^4) + 6*y^5 + 2*y^6;
%e n=5: R_4(y+y^2) = (y^2 + 6*y^3 + 20*y^4 + 40*y^5) + 46*y^6 + 28*y^7 + 7*y^8;
%e n=6: R_5(y+y^2) = (y^2 + 8*y^3 + 39*y^4 + 138*y^5 + 326*y^6) + 480*y^7 + 420*y^8 + 200*y^9 + 40*y^10;
%e where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n-1 nonzero terms.
%e ...
%e ALTERNATE GENERATING METHOD.
%e Let F^n(x) denote the n-th iteration of x+x^2 with F^0(x) = x.
%e Then row n of this triangle may be generated by the coefficients of x^k in G(F^n(x)), k=2..n, n>=2, where G(x) is the g.f. of A187119:
%e G(x) = x^2 - 2*x^3 + 4*x^4 - 12*x^5 + 36*x^6 - 140*x^7 + 519*x^8 - 2632*x^9 + 11776*x^10 - 82020*x^11 + 426990*x^12 +...
%e and satisfies: [x^(n+2)] G(F^n(x)) = 0 for n>0.
%e The table of coefficients in G(F^n(x)) begins:
%e G(x+x^2) : [1, 0, -1, -2, -2, -28, -37, -760, -1752,...];
%e G(F^2(x)): [1, 2, 0, -6, -18, -64, -284, -1694, -10640, ...];
%e G(F^3(x)): [1, 4, 7, 0, -46, -232, -1062, -5700, -36354, ...];
%e G(F^4(x)): [1, 6, 20, 40, 0, -480, -3369, -19988, -126200, ...];
%e G(F^5(x)): [1, 8, 39, 138, 326, 0, -6309, -56820, -417184, ...];
%e G(F^6(x)): [1, 10, 64, 318, 1258, 3492, 0, -100082, -1100188, ...];
%e G(F^7(x)): [1, 12, 95, 604, 3242, 14476, 46558, 0, -1859518, ...];
%e G(F^8(x)): [1, 14, 132, 1020, 6844, 40348, 202655, 744320, 0, ...];
%e of which this triangle forms the lower triangular portion.
%e ...
%e TRANSFORMATIONS OF SHIFTED DIAGONALS BY TRIANGLE A135080.
%e Given main diagonal = A135082 = [0,1,2,7,40,326,3492,46558,...],
%e the diagonals can be generated from each other as illustrated by:
%e _ A135080 * A135082 = A187116 = [0,1,4,20,138,1258,14476,202655,...];
%e _ A135080 * A187116 = A187117 = [0,1,6,39,318,3242,40348,598083,...];
%e _ A135080 * A187117 = [0,1,8,64,604,6844,92140,1450388,...],
%e where a leading zero is included in forming the vectors.
%e Related triangle A135080 begins:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 8, 7, 3, 1;
%e 50, 40, 15, 4, 1;
%e 436, 326, 112, 26, 5, 1;
%e 4912, 3492, 1128, 240, 40, 6, 1; ...
%e where column 1 of A135080 is the main diagonal in this triangle.
%o (PARI) {T(n,k)=local(Rn=y^2);for(m=2,n,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,k,y)}
%o (PARI) {T(n,k)=if(k>n|k<2,0,if(n==2,1,sum(j=k\2,k,binomial(j,k-j)*T(n-1,j))))}
%o /* Print the triangle: */
%o {for(n=2,12,for(k=2,n,print1(T(n,k),","));print(""))}
%Y Cf. diagonals: A135082, A187116, A187117; row sums: A187118.
%Y Cf. related triangles: A135080, A187005, A187120.
%Y Cf. A187119.
%K nonn,tabl
%O 2,3
%A _Paul D. Hanna_, Mar 07 2011
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