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

0,2

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).

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)

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 *)

Prog

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

Crossrefs

Apparently a bisection of A049779 and of A024862.

Cf. A180681, A332662.

Sequence in context: A179094 A373538 A176874 * A241765 A106956 A084671

Adjacent sequences: A331984 A331985 A331986 * A331988 A331989 A331990

Keyword

nonn

Author

Peter Luschny, Feb 19 2020

Status

approved