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A206294 Riordan array (1, x/(1-x)^3).

%I A206294 #17 Oct 07 2022 05:59:33
%S A206294 1,0,1,0,3,1,0,6,6,1,0,10,21,9,1,0,15,56,45,12,1,0,21,126,165,78,15,1,
%T A206294 0,28,252,495,364,120,18,1,0,36,462,1287,1365,680,171,21,1,0,45,792,
%U A206294 3003,4368,3060,1140,231,24,1
%N A206294 Riordan array (1, x/(1-x)^3).
%C A206294 The convolution triangle of the triangular numbers A000217. - _Peter Luschny_, Oct 07 2022
%H A206294 G. C. Greubel, <a href="/A206294/b206294.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A206294 Triangle T(n,k), read by rows, given by (0, 3, -1, 2/3, -1/6, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%F A206294 T(n,0) = 0^n, T(n,k) = C(n-1+2k, n-k) for k > 0.
%F A206294 T(n,n) = 1, T(k+1,k) = 3*k = A008585(k), T(k+2,k) = A081266(k).
%F A206294 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A052529(n), A052910(n) for x = 0, 1, 2 respectively.
%F A206294 G.f.: (1-x)^3/((1-x)^3-y*x).
%e A206294 Triangle begins:
%e A206294 1
%e A206294 0, 1
%e A206294 0, 3, 1
%e A206294 0, 6, 6, 1
%e A206294 0, 10, 21, 9, 1
%e A206294 0, 15, 56, 45, 12, 1
%e A206294 0, 21, 126, 165, 78, 15, 1
%e A206294 0, 28, 252, 495, 364, 120, 18, 1
%e A206294 0, 36, 462, 1287, 1365, 680, 171, 21, 1
%e A206294 0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1
%e A206294 0, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1
%e A206294 0, 66, 2002, 12870, 31824, 38760, 26324, 10626, 2600, 378, 30, 1
%p A206294 # Uses function PMatrix from A357368.
%p A206294 PMatrix(10, n -> n * (n + 1) / 2); # _Peter Luschny_, Oct 07 2022
%t A206294 Table[If[n == 0 && k == 0 , 1, Binomial[n - 1 + 2 k, n - k]], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 25 2017 *)
%o A206294 (PARI) {T(n,k)=polcoeff(1/(1-x+x*O(x^(n-k)))^(3*k),n-k)}
%o A206294 (PARI) {T(n,k)=polcoeff(polcoeff((1-x)^3/((1-x)^3-y*x +x*O(x^n)),n,x),k,y)}
%o A206294 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A206294 Cf. Columns: A000007, A000217 (triangular numbers), A000389, A000581, A001288, A010967..(+3)..A011000, A017714..(+3)..A017762.
%Y A206294 Row sums are A052529.
%Y A206294 Cf. A127893.
%K A206294 easy,nonn,tabl
%O A206294 0,5
%A A206294 _Philippe Deléham_, Feb 05 2012