A062749 Sixth column (r=5) of FS(3) staircase array A062745.

12, 43, 108, 228, 431, 753, 1239, 1944, 2934, 4287, 6094, 8460, 11505, 15365, 20193, 26160, 33456, 42291, 52896, 65524, 80451, 97977, 118427, 142152, 169530, 200967, 236898, 277788, 324133, 376461

Offset: 0

Author

Wolfdieter Lang, Jul 12 2001

Keywords

• nonn
• easy

Comments

In the Frey-Sellers reference this sequence is called {(n+3) over 5}_{2}, n >= 0.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Programs

Maple

seq(coeff(series((3*x^4-15*x^3+30*x^2-29*x+12)/(1-x)^6,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 30 2018

PARI

Vec((12 - 29*x + 30*x^2 - 15*x^3 + 3*x^4) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Oct 30 2018

Formula

a(n) = A062745(n+3, 5)= -3+binomial(n+4, 3)*(n^2+16*n+75)/20 = (n+1)*(n^4+24*n^3+221*n^2+894*n+1440)/5!.

G.f.: N(3;2, x)/(1-x)^6 with N(3;2, x)= 12-29*x+30*x^2-15*x^3+3*x^4, polynomial of the third row of A062746.

From Colin Barker, Oct 30 2018: (Start)

G.f.: (12 - 29*x + 30*x^2 - 15*x^3 + 3*x^4) / (1 - x)^6.

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>5.

(End)