OFFSET
0,3
COMMENTS
More generally, the generalized 4-dimensional figurate numbers are convolution of the sequence {1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...} with generalized polygonal numbers (A195152).
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).
FORMULA
G.f.: x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002419): (3*n - 1)*binomial(n + 2,3)/2, n = 0,+1,-3,+2,-4,+3,-5, ...
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(6*n^2+30*n+5-(2*n+5)*(-1)^n)/1536. - Luce ETIENNE, Nov 18 2017
MATHEMATICA
CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 51}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 5, 10, 26, 40, 80, 110, 190}, 52]
PROG
(PARI) x='x+O('x^99); concat(0, Vec(x*(1+4*x+x^2)/((1-x)^5*(1 + x)^4))) \\ Altug Alkan, Aug 15 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Aug 15 2017
STATUS
approved