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 A059594 Convolution triangle based on A008619 (positive integers repeated). 6
 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, 69, 85, 60, 27, 7, 1, 5, 40, 119, 176, 160, 92, 35, 8, 1, 5, 55, 189, 344, 376, 273, 133, 44, 9, 1, 6, 70, 294, 624, 820, 714, 434 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. 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). 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. The sequence of row sums is A006054(n+2). From Gary W. Adamson, Aug 14 2016: (Start) 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:   1, 1,  2,  2,  3,   3, ...   1, 2,  5,  8, 14,  20, ...   1, 3,  9, 19, 39,  69, ...   1, 4, 14, 36, 85, 176, ...   ... Row sums of the triangle = (1, 2, 5, 11, 25, 56, ...), the INVERT transform of (1, 1, 2, 2, 3, 3, ...). (End) LINKS FORMULA 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. G.f.for column m >= 0: ((x/((1-x^2)*(1-x)))^m)/((1-x^2)*(1-x)). 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 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 EXAMPLE {1}; {1,1}; {2,2,1}; {2,5,3,1}; ... Fourth row polynomial (n=3): p(3,x)= 2 + 5*x + 3*x^2 + x^3. MATHEMATICA 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 *) PROG (Maxima) 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 */ CROSSREFS Sequence in context: A325182 A215959 A209555 * A183760 A125678 A091562 Adjacent sequences:  A059591 A059592 A059593 * A059595 A059596 A059597 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Feb 02 2001 STATUS approved

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Last modified August 4 15:53 EDT 2021. Contains 346447 sequences. (Running on oeis4.)