A187120 - OEIS

%I #17 Nov 15 2024 18:03:21

%S 1,1,3,1,6,15,1,9,42,112,1,12,81,377,1128,1,15,132,855,4248,14373,1,

%T 18,195,1606,10758,58269,221952,1,21,270,2690,22416,159633,947117,

%U 4029915,1,24,357,4167,41340,359616,2750067,17848872,84135510,1,27,456,6097

%N Triangle, read by rows, where row n equals the coefficients of y^k in R_{n-1}(y+y^2) for k=3..n, where R_n(y) is the n-th row polynomial in y for n>=3 with R_3(y)=y^3.

%F 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.

%F Main diagonal equals column 2 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 2 of the m-th power of triangle A135080, with diagonal m=1 being the main diagonal.

%e Triangle begins:

%e 1;

%e 1, 3;

%e 1, 6, 15;

%e 1, 9, 42, 112;

%e 1, 12, 81, 377, 1128;

%e 1, 15, 132, 855, 4248, 14373;

%e 1, 18, 195, 1606, 10758, 58269, 221952;

%e 1, 21, 270, 2690, 22416, 159633, 947117, 4029915;

%e 1, 24, 357, 4167, 41340, 359616, 2750067, 17848872, 84135510;

%e 1, 27, 456, 6097, 70008, 715095, 6580260, 54178485, 383237040, 1985740905;

%e 1, 30, 567, 8540, 111258, 1301193, 13895408, 135965676, 1204443432, 9243654925, 52277994396; ...

%e in which rows can be generated as illustrated below.

%e Row polynomials R_n(y), n>=3, begin:

%e R_3(y) = y^3;

%e R_4(y) = y^3 + 3*y^4;

%e R_5(y) = y^3 + 6*y^4 + 15*y^5;

%e R_6(y) = y^3 + 9*y^4 + 42*y^5 + 112*y^6;

%e R_7(y) = y^3 + 12*y^4 + 81*y^5 + 377*y^6 + 1128*y^7; ...

%e where row n = coefficients of y^k in R_{n-1}(y+y^2) for k=3..n;

%e this method is illustrated by:

%e n=4: R_3(y+y^2) = (y^3 + 3*y^4) + 3*y^5 + y^6;

%e n=5: R_4(y+y^2) = (y^3 + 6*y^4 + 15*y^5) + 19*y^6 + 12*y^7 + 3*y^8;

%e 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; ...

%e where the n-th row polynomial R_n(y) equals R_{n-1}(y+y^2) truncated to the initial n-2 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-2](x)), k=3..n, n>=3, where G(x) is the g.f. of A187124:

%e 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 +...

%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, -3, -5, -12, -72, -333, -2568, -16782, ...];

%e G(F^2(x)): [1, 3, 0, -19, -72, -261, -1276, -8079, -58932, ...];

%e G(F^3(x)): [1, 6, 15, 0, -174, -1047, -5256, -29676, -202908, ...];

%e G(F^4(x)): [1, 9, 42, 112, 0, -2109, -17211, -112371, -753606, ...];

%e G(F^5(x)): [1, 12, 81, 377, 1128, 0, -31633, -324600, -2614344, ...];

%e G(F^6(x)): [1, 15, 132, 855, 4248, 14373, 0, -564081, -6957390, ...];

%e G(F^7(x)): [1, 18, 195, 1606, 10758, 58269, 221952, 0, -11639502,..];

%e G(F^8(x)): [1, 21, 270, 2690, 22416, 159633, 947117, 4029915, 0,...]; ...

%e of which this triangle forms the lower triangular portion.

%e ...

%e TRANSFORMATIONS OF SHIFTED DIAGONALS BY TRIANGLE A135080.

%e Given main diagonal = A135083 = [0,0,1,3,15,112,1128,14373,...],

%e the diagonals can be generated from each other as illustrated by:

%e _ A135080 * A135083 = A187121 = [0,0,1,6,42,377,4248,58269,...];

%e _ A135080 * A187121 = A187122 = [0,0,1,9,81,855,10758,159633,...];

%e _ A135080 * A187122 = [0,0,1,12,132,1606,22416,359616,...],

%e where two leading zeros are 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 2 of A135080 is the main diagonal in this triangle.

%o (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)}

%o (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))))}

%o /* Print the triangle: */

%o {for(n=3,13,for(k=3,n,print1(T(n,k),","));print(""))}

%Y Cf. diagonals: A135083, A187121, A187122; row sums: A187123.

%Y Cf. related triangles: A135080, A187005, A187115.

%Y Cf. A187124.

%K nonn,tabl

%O 3,3

%A _Paul D. Hanna_, Mar 08 2011