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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,3

LINKS

Table of n, a(n) for n=2..56.

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.

EXAMPLE

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.

PROG

(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(""))}

CROSSREFS

Cf. diagonals: A135082, A187116, A187117; row sums: A187118.

Cf. related triangles: A135080, A187005, A187120.

Cf. A187119.

Sequence in context: A223092 A071948 A193589 * A121722 A193591 A218842

Adjacent sequences:  A187112 A187113 A187114 * A187116 A187117 A187118

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 07 2011

STATUS

approved

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Last modified July 7 15:31 EDT 2020. Contains 335495 sequences. (Running on oeis4.)