A107880 - OEIS

A107880

Matrix square of triangle A107876; equals matrix product of triangles: A107876^2 = A107862^-1*A107870 = A107867^-1*A107873.

1, 2, 1, 3, 2, 1, 7, 5, 2, 1, 26, 19, 7, 2, 1, 141, 104, 37, 9, 2, 1, 1034, 766, 268, 61, 11, 2, 1, 9693, 7197, 2496, 550, 91, 13, 2, 1, 111522, 82910, 28612, 6195, 982, 127, 15, 2, 1, 1528112, 1136923, 391189, 83837, 12977, 1596, 169, 17, 2, 1, 24372513, 18141867, 6230646, 1326923, 202494, 24206, 2424, 217, 19, 2, 1

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OFFSET

0,2

COMMENTS

Column 0 is A107881. Column 1 is A107882. Column 3 equals A107883. Column 2 equals SHIFT_LEFT(A107877), where A107877 is column 1 of A107876.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(2+(k+j)*(k+j-1)/2-k*(k-1)/2).

EXAMPLE

G.f. for column 0:

1 = T(0,0)*(1-x)^2 + T(1,0)*x*(1-x)^2 + T(2,0)*x^2*(1-x)^3 + T(3,0)*x^3*(1-x)^5 + T(4,0)*x^4*(1-x)^8 + T(5,0)*x^5*(1-x)^12 +...

= 1*(1-x)^2 + 2*x*(1-x)^2 + 3*x^2*(1-x)^3 + 7*x^3*(1-x)^5 + 26*x^4*(1-x)^8 + 141*x^5*(1-x)^12 +...

G.f. for column 1:

1 = T(1,1)*(1-x)^2 + T(2,1)*x*(1-x)^3 + T(3,1)*x^2*(1-x)^5 + T(4,1)*x^3*(1-x)^8 + T(5,1)*x^4*(1-x)^12 + T(6,1)*x^5*(1-x)^17 +...

= 1*(1-x)^2 + 2*x*(1-x)^3 + 5*x^2*(1-x)^5 + 19*x^3*(1-x)^8 + 104*x^4*(1-x)^12 + 766*x^5*(1-x)^17 +...

Triangle T begins:

2,1;

3,2,1;

7,5,2,1;

26,19,7,2,1;

141,104,37,9,2,1;

1034,766,268,61,11,2,1;

9693,7197,2496,550,91,13,2,1;

111522,82910,28612,6195,982,127,15,2,1;

...

MATHEMATICA

max = 10;

A107862 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n - k], n - k], {n, 0, max}, {k, 0, max}];

A107867 = Table[Binomial[If[n < k, 0, n*(n-1)/2-k*(k-1)/2 + n-k+1], n - k], {n, 0, max}, {k, 0, max}];

T = MatrixPower[Inverse[A107862].A107867, 2];

Table[T[[n+1, k+1]], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 31 2024 *)

PROG

(PARI) {T(n, k)=polcoeff(1-sum(j=0, n-k-1, T(j+k, k)*x^j*(1-x+x*O(x^n))^(2+(k+j)*(k+j-1)/2-k*(k-1)/2)), n-k)}

CROSSREFS

Cf. A107862, A107870, A107873, A107867, A107876, A107884, A107887.

Sequence in context: A321155 A185624 A162387 * A102228 A141675 A248809

Adjacent sequences: A107877 A107878 A107879 * A107881 A107882 A107883

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jun 04 2005

STATUS

approved