A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).

1, 1, 1, 3, 6, 3, 13, 39, 39, 13, 75, 300, 450, 300, 75, 541, 2705, 5410, 5410, 2705, 541, 4683, 28098, 70245, 93660, 70245, 28098, 4683, 47293, 331051, 993153, 1655255, 1655255, 993153, 331051, 47293, 545835, 4366680, 15283380, 30566760, 38208450, 30566760, 15283380, 4366680, 545835

Offset: 0

Author

Peter Luschny, Apr 26 2023

Keywords

• nonn
• tabl

Examples

			[0]    1;
[1]    1,     1;
[2]    3,     6,     3;
[3]   13,    39,    39,    13;
[4]   75,   300,   450,   300,    75;
[5]  541,  2705,  5410,  5410,  2705,   541;
[6] 4683, 28098, 70245, 93660, 70245, 28098, 4683;
		

Crossrefs

Family of triangles: A055372 (m=0, Pascal), this sequence (m=1, Fubini), A362586 (m=2, Joffe), A362849 (m=3, A278073).

Cf. A000670 (column 0 and main diagonal), A216794 (row sums).

Programs

SageMath

def TransOrdPart(m, n) -> list[int]:
    @cached_function
    def P(m: int, n: int):
        R = PolynomialRing(ZZ, "x")
        if n == 0: return R(1)
        return R(sum(binomial(m * n, m * k) * P(m, n - k) * x
                 for k in range(1, n + 1)))
    T = P(m, n)
    def C(k) -> int:
        return sum(T[j] * binomial(n, k) for j in range(n + 1))
    return [C(k) for k in range(n+1)]
def A362585(n) -> list[int]: return TransOrdPart(1, n)
for n in range(6): print(A362585(n))