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 A030523 A convolution triangle of numbers obtained from A001792. 10
 1, 3, 1, 8, 6, 1, 20, 25, 9, 1, 48, 88, 51, 12, 1, 112, 280, 231, 86, 15, 1, 256, 832, 912, 476, 130, 18, 1, 576, 2352, 3276, 2241, 850, 183, 21, 1, 1280, 6400, 10976, 9424, 4645, 1380, 245, 24, 1, 2816, 16896, 34848, 36432, 22363, 8583, 2093, 316, 27, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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). 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 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 LINKS W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. W. Lang, First ten rows. FORMULA a(n, 1) = A001792(n-1). Row sums = A039717(n). 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 n or if k < 0. - Philippe Deléham, Feb 20 2013 Sum_{k=1..n} T(n,k)*2^(k-1) = A140766(n). -Philippe Deléham, Feb 20 2013 G.f.: (1-2*x)^2/((x^2-x)*y+(1-2*x)^2)-1. - Vladimir Kruchinin, Apr 28 2015 EXAMPLE {1}; {3,1}; {8,6,1}; {20,25,9,1}; {48,88,51,12,1}; ... (0, 3, -1/3, 4/3, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins: 1 0   1 0   3   1 0   8   6   1 0  20  25   9   1 0  48  88  51  12   1 ... - Philippe Deléham, Feb 20 2013 MATHEMATICA 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 *) CROSSREFS Cf. A057682 (alternating row sums). Sequence in context: A188939 A062196 A103247 * A125662 A123965 A124025 Adjacent sequences:  A030520 A030521 A030522 * A030524 A030525 A030526 KEYWORD easy,nonn,tabl AUTHOR STATUS approved

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