A354608 Number of quadruples (p_1, ..., p_4) of positive integers such that p_{i-1} <= p_i <= n^(i-1).

0, 1, 44, 541, 3236, 12885, 39656, 102249, 231736, 476121, 905620, 1618661, 2748604, 4471181, 7012656, 10658705, 15764016, 22762609, 32178876, 44639341, 60885140, 81785221, 108350264, 141747321, 183315176, 234580425, 297274276, 373350069, 465001516, 574681661

Offset: 0

Author

Alois P. Heinz, Jul 08 2022

Keywords

• nonn
• easy

Examples

			a(2) = 44: (1,1,1,1), (1,1,1,2), (1,1,1,3), (1,1,1,4), (1,1,1,5), (1,1,1,6), (1,1,1,7), (1,1,1,8), (1,1,2,2), (1,1,2,3), (1,1,2,4), (1,1,2,5), (1,1,2,6), (1,1,2,7), (1,1,2,8), (1,1,3,3), (1,1,3,4), (1,1,3,5), (1,1,3,6), (1,1,3,7), (1,1,3,8), (1,1,4,4), (1,1,4,5), (1,1,4,6), (1,1,4,7), (1,1,4,8), (1,2,2,2), (1,2,2,3), (1,2,2,4), (1,2,2,5), (1,2,2,6), (1,2,2,7), (1,2,2,8), (1,2,3,3), (1,2,3,4), (1,2,3,5), (1,2,3,6), (1,2,3,7), (1,2,3,8), (1,2,4,4), (1,2,4,5), (1,2,4,6), (1,2,4,7), (1,2,4,8).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Row n=4 of A355576.

Programs

Maple

a:= n-> (((((6*n-6)*n+3)*n+4)*n-3)*n+2)*n/6:
seq(a(n), n=0..30);

Formula

a(n) = (6*n^6-6*n^5+3*n^4+4*n^3-3*n^2+2*n)/6.

G.f.: x*(x^5+89*x^4+338*x^3+254*x^2+37*x+1)/(1-x)^7.