A054458 - OEIS

%I #26 Apr 03 2024 05:03:12

%S 1,3,1,7,6,1,17,23,9,1,41,76,48,12,1,99,233,204,82,15,1,239,682,765,

%T 428,125,18,1,577,1935,2649,1907,775,177,21,1,1393,5368,8680,7656,

%U 4010,1272,238,24,1,3363,14641,27312,28548,18358,7506,1946,308,27,1

%N Convolution triangle based on A001333(n), n >= 1.

%C In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.

%C The G.f. for the row polynomials p(n,x) (increasing powers of x) is LPell(z)/(1-x*z*LPell(z)) with LPell(z) given in 'Formula'.

%C Column sequences are A001333(n+1), A054459(n), A054460(n) for m=0..2.

%C Mirror image of triangle in A209696. - _Philippe Deléham_, Mar 24 2012

%C Subtriangle of the triangle given by (0, 3, -2/3, -1/3, 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_, Mar 25 2012

%C Riordan array ((1+x)/(1-2*x-x^2), (x+x^2)/(1-2*x-x^2)). - _Philippe Deléham_, Mar 25 2012

%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 a(n, m) := ((n-m+1)*a(n, m-1) + (2n-m)*a(n-1, m-1) + (n-1)*a(n-2, m-1))/(4*m), n >= m >= 1; a(n, 0)= A001333(n+1); a(n, m) := 0 if n<m.

%F G.f. for column m: LPell(x)*(x*LPell(x))^m, m >= 0, with LPell(x)= (1+x)/(1-2*x-x^2) = g.f. for A001333(n+1).

%F G.f.: (1+x)/(1-2*x-y*x-x^2-y*x^2). - _Philippe Deléham_, Mar 25 2012

%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3 and T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 25 2012

%F Sum_{k=0..n} T(n,k)*x^k = A040000(n), A001333(n+1), A055099(n), A126473(n), A126501(n), A126528(n) for x = -1, 0, 1, 2, 3, 4 respectively. - _Philippe Deléham_, Mar 25 2012

%e Fourth row polynomial (n=3): p(3,x)= 17+23*x+9*x^2+x^3.

%e Triangle begins :

%e   1

%e   3, 1

%e   7, 6, 1

%e   17, 23, 9, 1

%e   41, 76, 48, 12, 1

%e   99, 233, 204, 82, 15, 1

%e   239, 682, 765, 428, 125, 18, 1. - _Philippe Deléham_, Mar 25 2012

%e (0, 3, -2/3, -1/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :

%e   1

%e   0, 1

%e   0, 3, 1

%e   0, 7, 6, 1

%e   0, 17, 23, 9, 1

%e   0, 41, 76, 48, 12, 1

%e   0, 99, 233, 204, 82, 15, 1

%e   0, 239, 682, 765, 428, 125, 15, 1. - _Philippe Deléham_, Mar 25 2012

%Y Cf. A002203(n+1)/2. Row sums: A055099(n).

%Y Cf. A001333, A054459, A054460.

%Y Cf. A040000, A001333, A055099, A126473, A126501, A126528.

%K easy,nonn,tabl

%O 0,2

%A _Wolfdieter Lang_, Apr 26 2000