OFFSET
3,3
FORMULA
T(n,k) = Sum_{j=[k/2],k} C(j,k-j)*T(n-1,j) for n>=3, k=3..n, with T(n,3)=1 and T(n,k)=0 when k>n or k<3.
Main diagonal equals column 2 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 2 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.
EXAMPLE
Triangle begins:
1;
1, 3;
1, 6, 15;
1, 9, 42, 112;
1, 12, 81, 377, 1128;
1, 15, 132, 855, 4248, 14373;
1, 18, 195, 1606, 10758, 58269, 221952;
1, 21, 270, 2690, 22416, 159633, 947117, 4029915;
1, 24, 357, 4167, 41340, 359616, 2750067, 17848872, 84135510;
1, 27, 456, 6097, 70008, 715095, 6580260, 54178485, 383237040, 1985740905;
1, 30, 567, 8540, 111258, 1301193, 13895408, 135965676, 1204443432, 9243654925, 52277994396; ...
in which rows can be generated as illustrated below.
Row polynomials R_n(y), n>=3, begin:
R_3(y) = y^3;
R_4(y) = y^3 + 3*y^4;
R_5(y) = y^3 + 6*y^4 + 15*y^5;
R_6(y) = y^3 + 9*y^4 + 42*y^5 + 112*y^6;
R_7(y) = y^3 + 12*y^4 + 81*y^5 + 377*y^6 + 1128*y^7; ...
where row n = coefficients of y^k in R_{n-1}(y+y^2) for k=3..n;
this method is illustrated by:
n=4: R_3(y+y^2) = (y^3 + 3*y^4) + 3*y^5 + y^6;
n=5: R_4(y+y^2) = (y^3 + 6*y^4 + 15*y^5) + 19*y^6 + 12*y^7 + 3*y^8;
n=6: R_5(y+y^2) = (y^3 + 9*y^4 + 42*y^5 + 112*y^6) + 174*y^7 + 156*y^8 + 75*y^9 + 15*y^10; ...
where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n-2 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-2](x)), k=3..n, n>=3, where G(x) is the g.f. of A187124:
G(x) = x^3 - 3*x^4 + 6*x^5 - 18*x^6 + 48*x^7 - 195*x^8 + 549*x^9 - 3465*x^10 + 7452*x^11 - 112707*x^12 - 5994*x^13 - 6866904*x^14 +...
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, -3, -5, -12, -72, -333, -2568, -16782, ...];
G(F^2(x)): [1, 3, 0, -19, -72, -261, -1276, -8079, -58932, ...];
G(F^3(x)): [1, 6, 15, 0, -174, -1047, -5256, -29676, -202908, ...];
G(F^4(x)): [1, 9, 42, 112, 0, -2109, -17211, -112371, -753606, ...];
G(F^5(x)): [1, 12, 81, 377, 1128, 0, -31633, -324600, -2614344, ...];
G(F^6(x)): [1, 15, 132, 855, 4248, 14373, 0, -564081, -6957390, ...];
G(F^7(x)): [1, 18, 195, 1606, 10758, 58269, 221952, 0, -11639502,..];
G(F^8(x)): [1, 21, 270, 2690, 22416, 159633, 947117, 4029915, 0,...]; ...
of which this triangle forms the lower triangular portion.
...
TRANSFORMATIONS OF SHIFTED DIAGONALS BY TRIANGLE A135080.
Given main diagonal = A135083 = [0,0,1,3,15,112,1128,14373,...],
the diagonals can be generated from each other as illustrated by:
where two leading zeros are 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 2 of A135080 is the main diagonal in this triangle.
PROG
(PARI) {T(n, k)=local(Rn=y^3); for(m=3, n-1, Rn=subst(truncate(Rn), y, y+y^2+O(y^m))); polcoeff(Rn, k, y)}
(PARI) {T(n, k)=if(k>n|k<3, 0, if(n==3, 1, sum(j=k\2, k, binomial(j, k-j)*T(n-1, j))))}
/* Print the triangle: */
{for(n=3, 13, for(k=3, n, print1(T(n, k), ", ")); print(""))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 08 2011
STATUS
approved