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Four-column array read by rows: T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4 X 4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) and the vector (1,1,1,1).
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%I #23 Aug 11 2019 01:24:56

%S 1,1,1,1,4,11,14,6,35,75,70,24,204,540,570,210,1524,3618,3528,1224,

%T 9894,25050,25524,9144,69612,169932,168828,59364,467736,1165908,

%U 1175208,417672,3226524,7947084,7944648,2806416,21924672,54371568,54612456,19359144,150267840,371199864

%N Four-column array read by rows: T(n,k) for k=0..3 is the k-th component of the vector obtained by multiplying the n-th power of the 4 X 4 matrix (1,1,1,1; 7,3,1,0; 12,2,0,0; 6,0,0,0) and the vector (1,1,1,1).

%C (n+1)-st set of 4 terms = leftmost finite differences of sequences generated from 3rd degree polynomials having n-th row coefficients, (given n = 1,2,3...) For example, first row is (1 1 1 1) with a corresponding polynomial x^3 + x^2 + x + 1. (f(x),x = 1,2,3...) = 4, 15, 40, 85, 156...Leftmost term of the sequence = 4, with finite difference rows: 11, 25, 45, 71...; 14, 20, 26, 32...; and 6, 6, 6, 6. Thus leftmost terms of the sequence 4, 15, 40...and the finite difference rows are (4 11 14 6) which is the second row.

%C The matrix generator is discussed in A028246, while 2nd degree polynomial examples are A091140, A091141 and A091140. The first degree case is A095795.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12).

%F G.f. ( 1 +x +x^2 +x^3 +7*x^5 +10*x^6 +2*x^7 -5*x^8 +7*x^9 -10*x^10 -2*x^12 +6*x^13 -16*x^14 -24*x^11 ) / ( 1-4*x^4-24*x^8+30*x^12+12*x^16 ). - _R. J. Mathar_, Jun 20 2011

%F a(n) = +4*a(n-4) +24*a(n-8) -30*a(n-12) -12*a(n-16).

%e 3rd set of 4 terms = (35, 75, 70, 24) since M^2 * [1 1 1 1] = [35 75 70 24].

%e 1,1,1,1;

%e 4,11,14,6;

%e 35,75,70,24;

%e 204,540,570,210;

%e 1524,3618,3528,1224;

%e 9894,25050,25524,9144;

%p M := Matrix(4,4,[1,1,1,1,7,3,1,0,12,2,0,0,6,0,0,0]) ;

%p v := Vector(4,[1,1,1,1]) ;

%p for i from 0 to 20 do

%p Mpr := (M ^ i).v ;

%p for j from 1 to 4 do

%p printf("%d,", Mpr[j]) ;

%p end do;

%p end do; # _R. J. Mathar_, Jun 20 2011

%t LinearRecurrence[{0,0,0,4,0,0,0,24,0,0,0,-30,0,0,0,-12},{1,1,1,1,4,11,14,6,35,75,70,24,204,540,570,210},50] (* _Harvey P. Dale_, Feb 08 2013 *)

%o (PARI) Vec((1+x+x^2+x^3+7*x^5+10*x^6+2*x^7-5*x^8+7*x^9-10*x^10-2*x^12 +6*x^13-16*x^14-24*x^11) / (1-4*x^4-24*x^8+30*x^12+12*x^16)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 21 2011

%Y Cf. A028246, A091140, A091141, A091142, A095795, A053698.

%K nonn,tabf,easy

%O 0,5

%A _Gary W. Adamson_, Jun 06 2004

%E Name added by _R. J. Mathar_, several entries corrected by _Charles R Greathouse IV_, Jun 21 2011