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; ...

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