This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A059594 #27 Jun 04 2021 22:59:58
%S A059594 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 A059594 69,85,60,27,7,1,5,40,119,176,160,92,35,8,1,5,55,189,344,376,273,133,
%U A059594 44,9,1,6,70,294,624,820,714,434
%N A059594 Convolution triangle based on A008619 (positive integers repeated).
%C A059594 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 A059594 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 A059594 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 A059594 The sequence of row sums is A006054(n+2).
%C A059594 From _Gary W. Adamson_, Aug 14 2016: (Start)
%C A059594 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 A059594 1, 1, 2, 2, 3, 3, ...
%C A059594 1, 2, 5, 8, 14, 20, ...
%C A059594 1, 3, 9, 19, 39, 69, ...
%C A059594 1, 4, 14, 36, 85, 176, ...
%C A059594 ...
%C A059594 Row sums of the triangle = (1, 2, 5, 11, 25, 56, ...), the INVERT transform of (1, 1, 2, 2, 3, 3, ...). (End)
%F A059594 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 A059594 G.f.for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/((1-x^2)*(1-x)).
%F A059594 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 A059594 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 A059594 {1}; {1,1}; {2,2,1}; {2,5,3,1}; ...
%e A059594 Fourth row polynomial (n=3): p(3,x)= 2 + 5*x + 3*x^2 + x^3.
%t A059594 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 A059594 (Maxima)
%o A059594 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 A059594 nonn,easy,tabl
%O A059594 0,4
%A A059594 _Wolfdieter Lang_, Feb 02 2001