A060922 - OEIS

%I #16 Oct 19 2022 11:12:29

%S 1,3,1,4,6,1,7,17,9,1,11,38,39,12,1,18,80,120,70,15,1,29,158,315,280,

%T 110,18,1,47,303,753,905,545,159,21,1,76,566,1687,2568,2120,942,217,

%U 24,1,123,1039,3612,6666,7043,4311

%N Convolution triangle for Lucas numbers A000032(n+1), n >= 0.

%C In the language of Shapiro et al. (see A053121 for the reference) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. G.f. for row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) is (1+2*z)/(1-(1+x)*z-(1+2*x)*z^2).

%C Row sums give A060925. Column sequences (without leading zeros) are, for m=0..6: A000032(n+1)= A000204(n+1) (Lucas), A004799(n+1), A060929-33.

%C Bisection of this triangle gives triangles A060923 (even part) and A060924 (odd part).

%C For the m-th column sequence (without leading zeros) one has: a(n+m,m)= (pL1(m,n)*L(n+2)+pL2(m,n)*L(n+1))/(m!*5^m), m >= 0, with the Lucas numbers L(n)=A000032(n), n >= 0 and the row polynomials pL1(n,x) := sum(A061188(n,m)*x^n,m=0..n) and pL2(n,x) := sum(A061189(n,m)*x^m,m=0..n).

%C Riordan array ((1+2*x)/(1-x-x^2), x*(1+2*x)/(1-x-x^2)). - _Philippe Deléham_, Jan 21 2014

%C T is the convolution triangle of A000204 (see A357368). - _Peter Luschny_, Oct 19 2022

%F a(n, m)=((n-m+1)*a(n, m-1)+2*(2*n-m)*a(n-1, m-1)+4*(n-1)*a(n-2, m-1))/(5*m), n >= m >= 1, a(n, 0)= A000204(n+1)= A000032(n+1).

%F G.f. for m-th column: ((1+2*x)/(1-x-x^2))* ((x*(1+2*x))/(1-x-x^2))^m.

%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Jan 21 2014

%e p(2,x) = 4+6*x+x^2.

%e Triangle begins:

%e 1 ;

%e 3, 1;

%e 4, 6, 1;

%e 7, 17, 9, 1;

%e 11, 38, 39, 12, 1;

%e 18, 80, 120, 70, 15, 1;

%e 29, 158, 315, 280, 110, 18, 1;

%e 47, 303, 753, 905, 545, 159, 21, 1;

%p # Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.

%p PMatrix(10, A000204); # _Peter Luschny_, Oct 19 2022

%Y Cf. A000032.

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Apr 20 2001

%E Example improved by _Philippe Deléham_, Jan 21 2014