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%I #36 Aug 28 2019 16:41:04
%S 1,3,1,8,6,1,20,25,9,1,48,88,51,12,1,112,280,231,86,15,1,256,832,912,
%T 476,130,18,1,576,2352,3276,2241,850,183,21,1,1280,6400,10976,9424,
%U 4645,1380,245,24,1,2816,16896,34848,36432,22363,8583,2093,316,27,1
%N A convolution triangle of numbers obtained from A001792.
%C a(n,m) := s1p(3; n,m), a member of a sequence of unsigned triangles including s1p(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). Signed version: (-1)^(n-m)*a(n,m) := s1(3; n,m).
%C With offset 0, this is T(n,k) = Sum_{i=0..n} C(n,i)*C(i+k+1,2k+1). Binomial transform of A078812 (product of lower triangular matrices). - _Paul Barry_, Jun 22 2004
%C Subtriangle of the triangle T(n,k) given by (0, 3, -1/3, 4/3, 0, 0, 0, 0, 0, 0, 0, ... ) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 20 2013
%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H W. Lang, <a href="/A030523/a030523.txt">First ten rows. </a>
%F a(n, 1) = A001792(n-1).
%F Row sums = A039717(n).
%F a(n, m) = 2*(2*m+n-1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1. G.f. for m-th column: (x*(1-x)/(1-2*x)^2)^m.
%F T(n,k) = 4*T(n-1,k) - 4*T(n-2,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k > n or if k < 0. - _Philippe Deléham_, Feb 20 2013
%F Sum_{k=1..n} T(n,k)*2^(k-1) = A140766(n). -_Philippe Deléham_, Feb 20 2013
%F G.f.: (1-2*x)^2/((x^2-x)*y+(1-2*x)^2)-1. - _Vladimir Kruchinin_, Apr 28 2015
%e {1}; {3,1}; {8,6,1}; {20,25,9,1}; {48,88,51,12,1}; ...
%e (0, 3, -1/3, 4/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
%e 1
%e 0 1
%e 0 3 1
%e 0 8 6 1
%e 0 20 25 9 1
%e 0 48 88 51 12 1
%e ...
%e - _Philippe Deléham_, Feb 20 2013
%t a[n_, m_] := SeriesCoefficient[(1-2*x)^2/((x^2-x)*y + (1-2*x)^2) - 1, {x, 0, n}, {y, 0, m}]; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 28 2015, after _Vladimir Kruchinin_ *)
%Y Cf. A057682 (alternating row sums).
%K easy,nonn,tabl
%O 1,2
%A _Wolfdieter Lang_
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