A183065 Triangle defined by g.f.: Sum_{n>=0} (4*n)!/n!^4 * x^(2*n)*y^n/(1-x-x*y)^(4*n+1), read by rows.

1, 1, 1, 1, 26, 1, 1, 123, 123, 1, 1, 364, 3246, 364, 1, 1, 845, 25210, 25210, 845, 1, 1, 1686, 120135, 606500, 120135, 1686, 1, 1, 3031, 430941, 6082475, 6082475, 430941, 3031, 1, 1, 5048, 1277668, 38698856, 137915470, 38698856, 1277668, 5048, 1, 1, 7929

Offset

0,5

Comments

Compare the g.f. of this triangle with the g.f.s of triangles:

* A008459: Sum_{n>=0} (2n)!/n!^2 * x^(2n)*y^n/(1-x-xy)^(2n+1),

* A181543: Sum_{n>=0} (3n)!/n!^3 * x^(2n)*y^n/(1-x-xy)^(3n+1),

which have terms A008459(n,k) = C(n,k)^2 and A181543(n,k) = C(n,k)^3.

Links

Table of n, a(n) for n=0..46.

Example

G.f.: A(x,y) = 1/(1-x-xy) + 4!*x^2*y/(1-x-xy)^5 + (8!/2!^4)*x^4*y^2/(1-x-xy)^9 + (12!/3!^4)*x^6*y^3/(1-x-xy)^13 +...
Triangle begins:
1;
1, 1;
1, 26, 1;
1, 123, 123, 1;
1, 364, 3246, 364, 1;
1, 845, 25210, 25210, 845, 1;
1, 1686, 120135, 606500, 120135, 1686, 1;
1, 3031, 430941, 6082475, 6082475, 430941, 3031, 1;
1, 5048, 1277668, 38698856, 137915470, 38698856, 1277668, 5048, 1; ...

Prog

(PARI) {T(n, k)=polcoeff(polcoeff(sum(m=0, n, (4*m)!/m!^4*x^(2*m)*y^m/(1-x-x*y+x*O(x^n))^(4*m+1)), n, x), k, y)}

Crossrefs

Cf. A183066 (column 1), A183067 (row sums), A183068 (central terms).

Sequence in context: A225532 A177970 A225483 * A157630 A173583 A040687

Adjacent sequences: A183062 A183063 A183064 * A183066 A183067 A183068

Keyword

tabl,sign

Author

Paul D. Hanna, Dec 22 2010

Status

approved