PROG
(MAGMAMagma) [n^3+4*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013
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(MAGMAMagma) [n^3+4*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013
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G.f.: (1 +5x5*x -7x7*x^2 +x^3)/(1-x)^5.
a(0)=1, a(1)=10, a(2)=33, a(3)=76; for n>3, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Harvey P. Dale, Jan 24 2012
E.g.f.: (1 +9*x +7*x^2 +x^3)*exp(x). - G. C. Greubel, Aug 14 2019
(PARI) vector(40, n, n--; (n+1)^3+n*(n+1)) \\ G. C. Greubel, Aug 14 2019
(Sage) [(n+1)^3+n*(n+1) for n in (0..40)] # G. C. Greubel, Aug 14 2019
(GAP) List([0..40], n-> (n+1)^3+n*(n+1)); # G. C. Greubel, Aug 14 2019
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For q a prime power, a(q-1) = q^3 + q^2 - q is the number of pairs of commuting nilpotent 2*2 matrices with coefficients in GF(q). (Proof: the zero matrix commutes with all q^2 nilpotent matrices, there are q^2-1 non-zero nonzero nilpotent matrices, all conjugate, each commuting with q nilpotent matrices.) - Mark Wildon, Jun 20 2017
a(0)=1, a(1)=10, a(2)=33, a(3)=76; for n>3, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). [_- _Harvey P. Dale_, Jan 24 2012]
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