A107884 Matrix cube of triangle A107876; equals the product of triangular matrices: A107876^3 = A107862^-1*A107873.

1, 3, 1, 6, 3, 1, 16, 9, 3, 1, 63, 37, 12, 3, 1, 351, 210, 67, 15, 3, 1, 2609, 1575, 498, 106, 18, 3, 1, 24636, 14943, 4701, 975, 154, 21, 3, 1, 284631, 173109, 54298, 11100, 1689, 211, 24, 3, 1, 3909926, 2381814, 745734, 151148, 22518, 2688, 277, 27, 3, 1

Offset

0,2

Comments

Column 0 is A107885.

Column 1 is A107886.

Column 2 equals A107887.

Column 3 equals SHIFT_LEFT(A107878), where A107878 is column 2 of A107876.

Column 4 equals A107888.

Links

Table of n, a(n) for n=0..54.

Formula

G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(3 + (k+j)*(k+j-1)/2 - k*(k-1)/2).

Example

G.f. for column 0:
1 = T(0,0)*(1-x)^3 + T(1,0)*x*(1-x)^3 + T(2,0)*x^2*(1-x)^4 + T(3,0)*x^3*(1-x)^6 + T(4,0)*x^4*(1-x)^9 + T(5,0)*x^5*(1-x)^13 + ...
  = 1*(1-x)^3 + 3*x*(1-x)^3 + 6*x^2*(1-x)^4 + 16*x^3*(1-x)^6 + 63*x^4*(1-x)^9 + 351*x^5*(1-x)^13 + ...
G.f. for column 1:
1 = T(1,1)*(1-x)^3 + T(2,1)*x*(1-x)^4 + T(3,1)*x^2*(1-x)^6 + T(4,1)*x^3*(1-x)^9 + T(5,1)*x^4*(1-x)^13 + T(6,1)*x^5*(1-x)^18 + ...
  = 1*(1-x)^3 + 3*x*(1-x)^4 + 9*x^2*(1-x)^6 + 37*x^3*(1-x)^9 + 210*x^4*(1-x)^13 + 1575*x^5*(1-x)^18 + ...
Triangle begins:
       1;
       3,      1;
       6,      3,     1;
      16,      9,     3,     1;
      63,     37,    12,     3,    1;
     351,    210,    67,    15,    3,   1;
    2609,   1575,   498,   106,   18,   3,  1;
   24636,  14943,  4701,   975,  154,  21,  3, 1;
  284631, 173109, 54298, 11100, 1689, 211, 24, 3, 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, 3];
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))^(3+(k+j)*(k+j-1)/2-k*(k-1)/2)), n-k)}

Crossrefs

Cf. A107862, A107870, A107873, A107867, A107876, A107880, A107884, A107885, A107886, A107887, A107888.

Sequence in context: A133085 A039805 A094504 * A185628 A321194 A158822

Adjacent sequences: A107881 A107882 A107883 * A107885 A107886 A107887

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 04 2005

Status

approved