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 A209518 Triangle by rows, reversal of A104712. 1
 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, 28, 1, 9, 36, 84, 126, 126, 84, 36, 1, 10, 45, 120, 210, 252, 210, 120, 45, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The offset is chosen as "0" to match the generalized or compositional Bernoulli numbers. 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,...). 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,...]. 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". Row sums of the triangle = A000295 starting (1, 4, 11, 26, 57,...). LINKS Table of n, a(n) for n=0..54. Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568. FORMULA Doubly beheaded variant of Pascal's triangle in which two rightmost diagonals are deleted. 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 EXAMPLE First few rows of the triangle = 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, 28; 1, 9, 36, 84, 126, 126, 84, 36; 1, 10, 45, 120, 210, 252, 210, 120, 45; 1, 11, 55, 165, 330, 462, 462, 30, 165, 55; ... MATHEMATICA Table[Binomial[n+2, k+2], {n, 0, 9}, {k, n , 0, -1}] // Flatten (* Jean-François Alcover, Aug 08 2018 *) CROSSREFS Cf. A000295, A104721, A074909, A027641, A027642, A006568, A005669, A007318, A002411. Sequence in context: A086271 A345229 A080851 * A108285 A207619 A209694 Adjacent sequences: A209515 A209516 A209517 * A209519 A209520 A209521 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Mar 09 2012 STATUS approved

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Last modified July 17 18:09 EDT 2024. Contains 374377 sequences. (Running on oeis4.)