A187115 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.
1, 1, 2, 1, 4, 7, 1, 6, 20, 40, 1, 8, 39, 138, 326, 1, 10, 64, 318, 1258, 3492, 1, 12, 95, 604, 3242, 14476, 46558, 1, 14, 132, 1020, 6844, 40348, 202655, 744320, 1, 16, 175, 1590, 12750, 92140, 598083, 3354848, 13889080, 1, 18, 224, 2338, 21766, 185240, 1450388, 10337402, 64246776, 296459376
Offset: 2
Examples
Triangle begins:
1;
1, 2;
1, 4, 7;
1, 6, 20, 40;
1, 8, 39, 138, 326;
1, 10, 64, 318, 1258, 3492;
1, 12, 95, 604, 3242, 14476, 46558;
1, 14, 132, 1020, 6844, 40348, 202655, 744320;
1, 16, 175, 1590, 12750, 92140, 598083, 3354848, 13889080;
1, 18, 224, 2338, 21766, 185240, 1450388, 10337402, 64246776, 296459376;
1, 20, 279, 3288, 34818, 340112, 3097162, 26277556, 204706486, 1398909332, 7125938790; ...
in which rows can be generated as illustrated below.
Row polynomials R_n(y), n>=2, begin:
R_2(y) = y^2;
R_3(y) = y^2 + 2*y^3;
R_4(y) = y^2 + 4*y^3 + 7*y^4;
R_5(y) = y^2 + 6*y^3 + 20*y^4 + 40*y^5;
R_6(y) = y^2 + 8*y^3 + 39*y^4 + 138*y^5 + 326*y^6; ...
where row n = coefficients of y^k in R_{n-1}(y+y^2) for k=2..n;
this method is illustrated by:
n=3: R_2(y+y^2) = (y^2 + 2*y^3) + y^4;
n=4: R_3(y+y^2) = (y^2 + 4*y^3 + 7*y^4) + 6*y^5 + 2*y^6;
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;
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;
where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n-1 nonzero terms.
...
ALTERNATE GENERATING METHOD.
Let F^n(x) denote the n-th iteration of x+x^2 with F^0(x) = x.
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:
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 +...
and satisfies: [x^(n+2)] G(F^n(x)) = 0 for n>0.
The table of coefficients in G(F^n(x)) begins:
G(x+x^2) : [1, 0, -1, -2, -2, -28, -37, -760, -1752,...];
G(F^2(x)): [1, 2, 0, -6, -18, -64, -284, -1694, -10640, ...];
G(F^3(x)): [1, 4, 7, 0, -46, -232, -1062, -5700, -36354, ...];
G(F^4(x)): [1, 6, 20, 40, 0, -480, -3369, -19988, -126200, ...];
G(F^5(x)): [1, 8, 39, 138, 326, 0, -6309, -56820, -417184, ...];
G(F^6(x)): [1, 10, 64, 318, 1258, 3492, 0, -100082, -1100188, ...];
G(F^7(x)): [1, 12, 95, 604, 3242, 14476, 46558, 0, -1859518, ...];
G(F^8(x)): [1, 14, 132, 1020, 6844, 40348, 202655, 744320, 0, ...];
of which this triangle forms the lower triangular portion.
...
TRANSFORMATIONS OF SHIFTED DIAGONALS BY TRIANGLE A135080.
Given main diagonal = A135082 = [0,1,2,7,40,326,3492,46558,...],
the diagonals can be generated from each other as illustrated by:
_ A135080 * A135082 = A187116 = [0,1,4,20,138,1258,14476,202655,...];
_ A135080 * A187116 = A187117 = [0,1,6,39,318,3242,40348,598083,...];
_ A135080 * A187117 = [0,1,8,64,604,6844,92140,1450388,...],
where a leading zero is included in forming the vectors.
Related triangle A135080 begins:
1;
1, 1;
2, 2, 1;
8, 7, 3, 1;
50, 40, 15, 4, 1;
436, 326, 112, 26, 5, 1;
4912, 3492, 1128, 240, 40, 6, 1; ...
where column 1 of A135080 is the main diagonal in this triangle.
Crossrefs
Programs
-
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)} -
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))))} /* Print the triangle: */ {for(n=2,12,for(k=2,n,print1(T(n,k),","));print(""))}
Formula
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.
Main diagonal equals column 1 of triangle A135080, which transforms diagonals in the table of coefficients of the iterations of x+x^2.
Triangle A135080 also transforms diagonals in this triangle into each other.
Diagonal m of this triangle equals column 1 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.