cp's OEIS Frontend

A209518 Triangle by rows, reversal of A104712.

%I A209518 #24 Jul 25 2024 06:53:33
%S A209518 1,1,3,1,4,6,1,5,10,10,1,6,15,20,15,1,7,21,35,35,21,1,8,28,56,70,56,
%T A209518 28,1,9,36,84,126,126,84,36,1,10,45,120,210,252,210,120,45,1,11,55,
%U A209518 165,330,462,462,330,165,55
%N A209518 Triangle by rows, reversal of A104712.
%C A209518 The offset is chosen as "0" to match the generalized or compositional Bernoulli numbers.
%C A209518 Following [Blandin and Diaz], we can generalize a subset of Bernoulli numbers to comply with the origin of the triangle (the Pascal matrix A007318 beheaded once: (A074909), twice: (this triangle), and so on...); and a corresponding Bernoulli sequence that equals the inverse of the triangle, extracting the left border. This procedure done with A074909 results in The Bernoulli numbers (A027641/A026642) starting (1, -1/2, 1/6,...). Done with this triangle we obtain A006568/A006569: (1, -1/3, 1/18, 1/90,...).
%C A209518 A generalized algebraic property of the subset of such triangles and compositional Bernoulli numbers is that the triangle M * [corresponding Bernoulli sequence considered as a vector, V] = [1, 0, 0, 0,...].
%C A209518 The infinite set of generalized Bernoulli number sequences thus generated from variants of Pascal's triangle begins: [(1, -1/2, 1/6,...); (1, -1/3, 1/18,...); (1, -1/4, 1/40,...); (1, -1/5, 1/75,...); where the third term denominators = A002411 (1, 6, 18, 40, 75,...) after the "1".
%C A209518 Row sums of the triangle = A000295 starting (1, 4, 11, 26, 57,...).
%H A209518 Hector Blandin and Rafael Diaz, <a href="http://arxiv.org/abs/0708.0809">Compositional Bernoulli numbers</a>, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568.
%F A209518 Doubly beheaded variant of Pascal's triangle in which two rightmost diagonals are deleted.
%F A209518 T(n,k)=T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-3*T(n-2,k-2)+T(n-3,k-2)+T(n-3,k-3), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - _Philippe Deléham_, Jan 11 2014
%e A209518 First few rows of the triangle =
%e A209518 1;
%e A209518 1, 3;
%e A209518 1, 4, 6;
%e A209518 1, 5, 10, 10;
%e A209518 1, 6, 15, 20, 15;
%e A209518 1, 7, 21, 35, 35, 21;
%e A209518 1, 8, 28, 56, 70, 56, 28;
%e A209518 1, 9, 36, 84, 126, 126, 84, 36;
%e A209518 1, 10, 45, 120, 210, 252, 210, 120, 45;
%e A209518 1, 11, 55, 165, 330, 462, 462, 30, 165, 55;
%e A209518 ...
%t A209518 Table[Binomial[n+2, k+2], {n, 0, 9}, {k, n , 0, -1}] // Flatten (* _Jean-François Alcover_, Aug 08 2018 *)
%Y A209518 Cf. A000295, A104721, A074909, A027641, A027642, A006568, A006569, A007318, A002411.
%K A209518 nonn,tabl
%O A209518 0,3
%A A209518 _Gary W. Adamson_, Mar 09 2012