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%I #71 Sep 08 2022 08:45:13
%S 1,2,1,6,3,1,20,10,4,1,70,35,15,5,1,252,126,56,21,6,1,924,462,210,84,
%T 28,7,1,3432,1716,792,330,120,36,8,1,12870,6435,3003,1287,495,165,45,
%U 9,1,48620,24310,11440,5005,2002,715,220,55,10,1,184756,92378,43758,19448,8008,3003,1001,286,66,11,1
%N Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.
%C First column is C(2*n,n) or A000984. Central coefficients are C(3*n,n) or A005809. - _Paul Barry_, Oct 14 2009
%C T(n,k) = A046899(n,n-k), k = 0..n-1. - _Reinhard Zumkeller_, Jul 27 2012
%C From _Peter Bala_, Nov 03 2015: (Start)
%C Viewed as the square array [binomial (2*n + k, n + k)]n,k>=0 this is the generalized Riordan array ( 1/sqrt(1 - 4*x),c(x) ) in the sense of the Bala link, where c(x) is the o.g.f. for A000108.
%C The square array factorizes as ( 1/(2 - c(x)),x*c(x) ) * ( 1/(1 - x),1/(1 - x) ), which equals the matrix product of A100100 with the square Pascal matrix [binomial (n + k,k)]n,k>=0. See the example below. (End)
%H Reinhard Zumkeller, <a href="/A092392/b092392.txt">Rows n=0..149 of triangle, flattened</a>
%H P. Bala, <a href="/A260492/a260492.pdf">Notes on generalized Riordan arrays</a>
%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry1/barry242.html">On the Central Coefficients of Riordan Matrices</a>, Journal of Integer Sequences, 16 (2013), #13.5.1.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H Ik-Pyo Kim, Michael J. Tsatsomeros, <a href="https://arxiv.org/abs/1707.06590">Inverse Relations in Shapiro's Open Questions</a>, arXiv:1707.06590 [math.CO], 2017. See p. 7.
%F As a number triangle, this is T(n, k) = if(k <= n, C(2*n - k, n), 0). Its row sums are C(2*n + 1, n + 1) = A001700. Its diagonal sums are A176287. - _Paul Barry_, Apr 23 2005
%F G.f. of column k: 2^k/[sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))^k].
%F As a number triangle, this is the Riordan array (1/sqrt(1 - 4*x), x*c(x)), c(x) the g.f. of A000108. - _Paul Barry_, Jun 24 2005
%F G.f.: A(x,y)=1/sqrt(1 - 4*x)/(1-y*x*C(x)), where C(x) is g.f. of Catalan numbers. - _Vladimir Kruchinin_, Mar 19 2016
%e From _Paul Barry_, Oct 14 2009: (Start)
%e Triangle begins
%e 1,
%e 2, 1,
%e 6, 3, 1,
%e 20, 10, 4, 1,
%e 70, 35, 15, 5, 1,
%e 252, 126, 56, 21, 6, 1,
%e 924, 462, 210, 84, 28, 7, 1,
%e 3432, 1716, 792, 330, 120, 36, 8, 1
%e Production array is
%e 2, 1,
%e 2, 1, 1,
%e 2, 1, 1, 1,
%e 2, 1, 1, 1, 1,
%e 2, 1, 1, 1, 1, 1,
%e 2, 1, 1, 1, 1, 1, 1,
%e 2, 1, 1, 1, 1, 1, 1, 1,
%e 2, 1, 1, 1, 1, 1, 1, 1, 1,
%e 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
%e As a square array = A100100 * square Pascal matrix:
%e /1 1 1 1 ...\ / 1 \/1 1 1 1 ...\
%e |2 3 4 5 ...| | 1 1 ||1 2 3 4 ...|
%e |6 10 15 21 ...| = | 3 2 1 ||1 3 6 10 ...|
%e |20 35 56 84 ...| |10 6 3 1 ||1 4 10 20 ...|
%e |70 ... | |35 ... ||1 ... |
%e - _Peter Bala_, Nov 03 2015
%p A092392 := proc(n,k)
%p binomial(2*n-k,n-k) ;
%p end proc:
%p seq(seq(A092392(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Feb 06 2015
%t Table[Binomial[2 n - k, n], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Mar 19 2016 *)
%o (Haskell)
%o a092392 n k = a092392_tabl !! (n-1) !! (k-1)
%o a092392_row n = a092392_tabl !! (n-1)
%o a092392_tabl = map reverse a046899_tabl
%o -- _Reinhard Zumkeller_, Jul 27 2012
%o (Maxima)
%o C(x):=(1-sqrt(1-4*x))/2;
%o A(x,y):=(1/sqrt(1-4*x))/(1-y*C(x));
%o taylor(A(x,y),y,0,10,x,0,10); /* _Vladimir Kruchinin_, Mar 19 2016 */
%o (PARI) for(n=0,10, for(k=0,n, print1(binomial(2*n - k,n), ", "))) \\ _G. C. Greubel_, Nov 22 2017
%o (Magma) /* As a triangle */ [[Binomial(2*n-k, n): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 22 2017
%Y Rows 0-14 are A000984, A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.
%Y Columns are A000217, A000292, A000332, A000389, A000579.
%Y Diagonals are A005809, A045721, A025174, A004319, A013698, A003408.
%Y Cf. A100100.
%K nonn,tabl,easy
%O 0,2
%A _Ralf Stephan_, Mar 21 2004
%E Diagonal sums comment corrected by _Paul Barry_, Apr 14 2010
%E Offset corrected by _R. J. Mathar_, Feb 08 2013
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