|
%I #16 Sep 08 2022 08:45:28
%S 1,1,0,1,0,1,1,0,3,1,1,0,6,4,3,1,0,10,10,15,6,1,0,15,20,45,36,15,1,0,
%T 21,35,105,126,105,36,1,0,28,56,210,336,420,288,91,1,0,36,84,378,756,
%U 1260,1296,819,232,1,0,45,120,630,1512,3150,4320,4095,2320,603
%N Triangle read by rows: T(n,k) = binomial(n,k)*r(k), where r(k) are the Riordan numbers (r(k) = A005043(k); 0 <= k <= n).
%C Row sums = Catalan numbers, A000108: (1, 1, 2, 5, 14, 42...); e.g., sum of row 4 terms = A000108(4) = 14 = (1 + 0 + 6 + 4 + 3). A005043 is the inverse binomial transform of the Catalan numbers.
%H G. C. Greubel, <a href="/A124926/b124926.txt">Rows n = 0..100 of triangle, flattened</a>
%H Chao-Jen Wang, <a href="http://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf">Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords</a>, Dissertation, Brandeis University, 2011.
%e First few rows of the triangle:
%e 1;
%e 1, 0;
%e 1, 0, 1;
%e 1, 0, 3, 1;
%e 1, 0, 6, 4, 3;
%e 1, 0, 10, 10, 15, 6;
%e 1, 0, 15, 20, 45, 36, 15;
%e ...
%p r:=n->(1/(n+1))*sum((-1)^i*binomial(n+1,i)*binomial(2*n-2*i,n-i),i=0..n): T:=(n,k)->r(k)*binomial(n,k): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t T[n_, k_]:= T[n, k]= Binomial[n, k]*Sum[(-1)^j*Binomial[k+1, j]* Binomial[2*(k-j), k-j], {j,0,k}]/(k+1); Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 19 2019 *)
%o (PARI) T(n,k) = b=binomial; b(n,k)*sum(j=0,k, (-1)^j*b(k+1,j)*b(2*(k-j), k-j))/(k+1); \\ _G. C. Greubel_, Nov 19 2019
%o (Magma) B:=Binomial; [B(n,k)*(&+[(-1)^j*B(k+1,j)*B(2*(k-j), k-j): j in [0..k]])/(k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 19 2019
%o (Sage) b=binomial; [[b(n,k)*sum((-1)^j*b(k+1,j)*b(2*(k-j), k-j) for j in (0..k))/(k+1) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 19 2019
%o (GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> B(n,k)* Sum([0..k], j-> (-1)^j*B(k+1,j)*B(2*(k-j), k-j))/(k+1) ))); # _G. C. Greubel_, Nov 19 2019
%Y Cf. A000108, A005043.
%K nonn,tabl
%O 0,9
%A _Gary W. Adamson_, Nov 12 2006
%E Edited by _N. J. A. Sloane_, Nov 29 2006
|
