A304725 a(n) = n^4 + 8*n^3 + 20*n^2 + 16*n + 2.

2, 47, 194, 527, 1154, 2207, 3842, 6239, 9602, 14159, 20162, 27887, 37634, 49727, 64514, 82367, 103682, 128879, 158402, 192719, 232322, 277727, 329474, 388127, 454274, 528527, 611522, 703919, 806402, 919679, 1044482, 1181567, 1331714, 1495727, 1674434, 1868687

Offset: 0

Author

Vincenzo Librandi, May 30 2018

Keywords

• nonn
• easy

Links

• Rigoberto Florez, Robinson A. Higuita and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.5 (see formula for B_4(x) on page 4).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A008865.

Fourth column of the array in A298675 (without -1).

Fifth column of the array in A299741.

Programs

Magma

[n^4+8*n^3+20*n^2+16*n+2: n in [0..40]];

Mathematica

Table[n^4 + 8 n^3 + 20 n^2 + 16 n + 2, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{2,47,194,527,1154},50] (* Harvey P. Dale, Jan 23 2025 *)

Formula

G.f.: (2 + 37*x - 21*x^2 + 7*x^3 - x^4)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = A008865(n+2)^2 - 2. Therefore, a(n) is a member of A008865.