A331987 a(n) = ((n + 1) - 9*(n + 1)^2 + 8*(n + 1)^3)/6.

0, 5, 23, 62, 130, 235, 385, 588, 852, 1185, 1595, 2090, 2678, 3367, 4165, 5080, 6120, 7293, 8607, 10070, 11690, 13475, 15433, 17572, 19900, 22425, 25155, 28098, 31262, 34655, 38285, 42160, 46288, 50677, 55335, 60270, 65490, 71003, 76817, 82940, 89380, 96145

Offset: 0

Author

Peter Luschny, Feb 19 2020

Keywords

• nonn

Comments

The start values of the partial rows on the main diagonal of A332662 in the representation in the example section.

Apparently the sum of the hook lengths over the partitions of 2*n + 1 with exactly 2 parts (cf. A180681).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Apparently a bisection of A049779 and of A024862.

Cf. A180681, A332662.

Programs

Magma

[n*(n+1)*(8*n+7)/6: n in [0..50]]; // G. C. Greubel, Apr 19 2023

Maple

a := n -> ((n+1) - 9*(n+1)^2 + 8*(n+1)^3)/6: seq(a(n), n=0..41);
gf := (x*(3*x + 5))/(x - 1)^4: ser := series(gf, x, 44):
seq(coeff(ser, x, n), n=0..41);

Mathematica

LinearRecurrence[{4,-6,4,-1}, {0,5,23,62}, 42]
Table[(n-9n^2+8n^3)/6,{n,50}] (* Harvey P. Dale, Apr 11 2024 *)

SageMath

def A331987(n): return n*(n+1)*(8*n+7)/6
[A331987(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

Formula

a(n) = [x^n] (x*(5 + 3*x)/(1 - x)^4).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2).

From G. C. Greubel, Apr 19 2023: (Start)

a(n) = 3*binomial(n+1,1) - 11*binomial(n+2,2) + 8*binomial(n+3,3).

a(n) = n*binomial(8*n+8,2)/24.

a(n) = n*(n+1)*(8*n+7)/6.

E.g.f.: (1/6)*x*(30 + 39*x + 8*x^2)*exp(x). (End)