A030523 A convolution triangle of numbers obtained from A001792.

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

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

Table of n, a(n) for n=1..55.

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<m; a(n, 0) := 0; a(1, 1)=1. G.f. for m-th column: (x*(1-x)/(1-2*x)^2)^m.

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

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 * A207815 A123965 A125662

Adjacent sequences: A030520 A030521 A030522 * A030524 A030525 A030526

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Status

approved