A257890 Expansion of the g.f. (x^2-x+1)*(x^2-3*x+3)/(x-1)^6.

3, 12, 34, 80, 166, 314, 553, 920, 1461, 2232, 3300, 4744, 6656, 9142, 12323, 16336, 21335, 27492, 34998, 44064, 54922, 67826, 83053, 100904, 121705, 145808, 173592, 205464, 241860, 283246, 330119, 383008, 442475, 509116, 583562, 666480, 758574, 860586

Offset: 0

Author

R. J. Mathar, May 12 2015

Keywords

• nonn
• easy

Comments

Absolute values of the 5th column of A220074.

Convolution of A000124 and the sequence 3, 6, 10, 15 (the triangular numbers A000217 without the first two entries).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000124, A000217, A220074.

Programs

Magma

[(n+1)*(n^4+14*n^3+91*n^2+254*n+360)/120: n in [0..40]]; // Vincenzo Librandi, May 12 2015

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {3, 12, 34, 80, 166, 314}, 50] (* Vincenzo Librandi, May 12 2015 *)

PARI

Vec((x^2-x+1)*(x^2-3*x+3)/(x-1)^6 + O(x^50)) \\ Michel Marcus, Jan 28 2016

Formula

G.f.: (x^2-x+1)*(x^2-3*x+3)/(x-1)^6.

a(n) = A000292(n+1) + (n+1) + A000389(n+5).

a(n) = (n+1)*(n^4 +14*n^3 +91*n^2 +254*n +360)/120.

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) for n>6. - Wesley Ivan Hurt, Jan 27 2016

E.g.f.: (360 + 1080*x + 780*x^2 + 220*x^3 + 25*x^4 + x^5)*exp(x)/120. - G. C. Greubel, Nov 24 2017