%I #27 Jun 04 2021 22:59:58
%S 1,1,1,2,2,1,2,5,3,1,3,8,9,4,1,3,14,19,14,5,1,4,20,39,36,20,6,1,4,30,
%T 69,85,60,27,7,1,5,40,119,176,160,92,35,8,1,5,55,189,344,376,273,133,
%U 44,9,1,6,70,294,624,820,714,434
%N Convolution triangle based on A008619 (positive integers repeated).
%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) = Sum_{m=0..n} a(n,m)*x^m is 1/((1-z^2)*(1-z)-x*z).
%C The column sequences are A008619(n); A006918(n); A038163(n-2), n >= 2; A038164(n-3), n >= 3; A038165(n-4), n >= 4; A038166(n-5), n >= 5; A059595(n-6), n >= 6; A059596(n-7), n >= 7; A059597(n-8), n >= 8; A059598(n-9), n >= 9; A059625(n-10), n >= 10 for m=0..10.
%C The sequence of row sums is A006054(n+2).
%C From _Gary W. Adamson_, Aug 14 2016: (Start)
%C The sequence can be generated by extracting the descending antidiagonals of an array formed by taking powers of the natural integers with repeats, (1, 1, 2, 2, 3, 3, ...), as follows:
%C 1, 1, 2, 2, 3, 3, ...
%C 1, 2, 5, 8, 14, 20, ...
%C 1, 3, 9, 19, 39, 69, ...
%C 1, 4, 14, 36, 85, 176, ...
%C ...
%C Row sums of the triangle = (1, 2, 5, 11, 25, 56, ...), the INVERT transform of (1, 1, 2, 2, 3, 3, ...). (End)
%F a(n, m) := a(n-1, m) + (-(n-m+1)*a(n, m-1) + 3*(n+2*m)*a(n-1, m-1))/(8*m), n >= m >= 1; a(n, 0) := floor((n+2)/2) = A008619(n), n >= 0; a(n, m) := 0 if n < m.
%F G.f.for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/((1-x^2)*(1-x)).
%F T(n,m) = Sum_{k=0..n-m} (Sum_{j=0..k} binomial(j, n-m-3*k+2*j)*(-1)^(j-k)*binomial(k,j))*binomial(m+k,m). - _Vladimir Kruchinin_, Dec 14 2011
%F Recurrence: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-3,k) with T(0,0) = 1. - _Philippe Deléham_, Feb 23 2012
%e {1}; {1,1}; {2,2,1}; {2,5,3,1}; ...
%e Fourth row polynomial (n=3): p(3,x)= 2 + 5*x + 3*x^2 + x^3.
%t t[n_, m_] := Sum[Sum[Binomial[j, n-m-3*k+2*j]*(-1)^(j-k)*Binomial[k, j], {j, 0, k}]*Binomial[m+k, m], {k, 0, n-m}]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, May 27 2013, after _Vladimir Kruchinin_ *)
%o (Maxima)
%o T(n,m):=sum((sum(binomial(j,n-m-3*k+2*j)*(-1)^(j-k)*binomial(k,j),j,0,k)) *binomial(m+k,m),k,0,n-m); /* _Vladimir Kruchinin_, Dec 14 2011 */
%K nonn,easy,tabl
%O 0,4
%A _Wolfdieter Lang_, Feb 02 2001