%I #26 Oct 09 2019 03:56:24

%S 1,4,1,13,8,1,40,42,12,1,121,184,87,16,1,364,731,496,148,20,1,1093,

%T 2736,2454,1040,225,24,1,3280,9844,11064,6170,1880,318,28,1,9841,

%U 34448,46738,32624,13015,3080,427,32,1,29524,118101,188208,158724,79044,24381,4704

%N Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

%C Row sums are A116415. Diagonal sums are A007070. First column is A003462(n+1). Product of A007318 and A116412.

%C Subtriangle of triangle given by (0, 4, -3/4, 3/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jan 18 2012

%H Michael De Vlieger, <a href="/A116414/b116414.txt">Table of n, a(n) for n = 0..11324</a> (rows 0 <= n <= 150)

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.

%F Riordan array (1/(1-4x+3x^2), x/(1-4x+3x^2)); number triangle T(n,k) = Sum_{j=0..n} binomial(n-j,k)*binomial(k+j,j)*3^j.

%F T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(2,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Oct 31 2013

%F G.f.: (1-4*x+3*x^2)/(1-4*x+3*x^2-x*y). - _Philippe Deléham_, Oct 31 2013

%F From _Peter Bala_, Oct 07 2019: (Start)

%F O.g.f.: 1/(1 - 4*x + 3*x^2 - x*y) = 1 + (4 + y)*x + (13 + 8*y + y^2)*x^2 + ....

%F Recurrence for row polynomials: R(n,y) = (4 + y)*R(n-1,y) - 3*R(n-2,y) with R(0,y) = 1 and R(1,y) = 4 + y.

%F The row reverse polynomial y^n*R(n,1/y) is equal to the numerator polynomial of the finite continued fraction 1 + y/(1 + 3*y/(1 + ... + y/(1 + 3*y/(1)))) (with 2*n partial numerators). Cf. A110441. (End)

%e Triangle begins

%e 1;

%e 4, 1;

%e 13, 8, 1;

%e 40, 42, 12, 1;

%e 121, 184, 87, 16, 1;

%e 364, 731, 496, 148, 20, 1;

%e Triangle T(n,k), 0 <= k <= n, given by (0, 4, -3/4, 3/4, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:

%e 1;

%e 0, 1;

%e 0, 4, 1;

%e 0, 13, 8, 1;

%e 0, 40, 42, 12, 1;

%e 0, 121, 184, 87, 16, 1;

%e 0, 364, 731, 496, 148, 20, 1;

%e ... - _Philippe Deléham_, Jan 18 2012

%t With[{n = 10}, DeleteCases[#, 0] & /@ Rest@ CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 4 x + 3 x^2 - x y), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* _Michael De Vlieger_, Apr 25 2018 *)

%Y Cf. A003462, A116412, A110441.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Feb 13 2006