cp's OEIS Frontend

A129533 Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.

%I A129533 #45 Sep 01 2025 11:25:27
%S A129533 0,0,0,0,1,0,0,3,3,0,0,6,12,6,0,0,10,30,30,10,0,0,15,60,90,60,15,0,0,
%T A129533 21,105,210,210,105,21,0,0,28,168,420,560,420,168,28,0,0,36,252,756,
%U A129533 1260,1260,756,252,36,0,0,45,360,1260,2520,3150,2520,1260,360,45,0,0,55,495
%N A129533 Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
%C A129533 Previous name was: Triangle read by rows: T(n,k)=derivative of the q-binomial coefficient [n,k] evaluated at q=1 (0<=k<=n). - _N. J. A. Sloane_, Jan 06 2016
%C A129533 For example, T(5,2)=30 because [5,2] = q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1 with derivative 6q^5 + 5q^4 + 8q^3 + 6q^2 + 4q + 1, having value 30 at q=1. - _Emeric Deutsch_, Apr 22 2007
%C A129533 Sum of entries in n-th antidiagonal = n(n-1)2^(n-3) = A001788(n-1).
%C A129533 T(n,k) = A094305(n-2, k-1) for n >= 2, k >= 1.
%C A129533 T(n,k) is total number of pips on a set of generalized linear dominoes with n cells (rather than two) and with the number of pips in each cell running from 0 to k (rather than 6). T(2,6) = 168 gives the total number of pips on a standard set of dominoes. We regard a generalized linear domino with n cells and up to k pips per cell as an ordered n-tuple [i_1, i_2, ..., i_n] with 0 <= i_1 <= i_2 <=  ... <= i_n <= k. - _Alan Shore_ and _N. J. A. Sloane_, Jan 06 2016
%C A129533 T(n,k) can also be written more symmetrically as the trinomial coefficient (n+k; n-1, k-1, 2). - _N. J. A. Sloane_, Jan 06 2016
%C A129533 As a triangle read by rows, T(n,k) is the total number of inversions over all length n binary words having exactly k 1's. T(n,k) is also the total area above all North East lattice paths from the origin to the point (k,n-k). - _Geoffrey Critzer_, Mar 22 2018
%D A129533 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.
%H A129533 Cemil Karaçam and Alper Vural, <a href="https://doi.org/10.21136/MB.2025.0084-24">Enumerating 2D and 3D lattice paths with arbitrary steps</a>, Mathematica Bohemica, pp. 1-13 (2025). See p. 5.
%H A129533 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>.
%F A129533 T(n,k) = (1/2)*k*(k+1)*binomial(n,k+1).
%F A129533 G.f.: G(q,z) = qz^2/(1-z-qz)^3.
%e A129533 Array begins:
%e A129533 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... (A000004)
%e A129533 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ... (A000217)
%e A129533 0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, ... (A027480)
%e A129533 0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, ... (A033487)
%e A129533 0, 10, 60, 210, 560, 1260, 2520, 4620, 7920, 12870, ... (A266732)
%e A129533 0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, ... (A240440)
%e A129533 0, 21, 168, 756, 2520, 6930, 16632, 36036, ... (A266733)
%e A129533 ...
%e A129533 If regarded as a triangle, this begins:
%e A129533   0;
%e A129533   0,  0;
%e A129533   0,  1,  0;
%e A129533   0,  3,  3,  0;
%e A129533   0,  6, 12,  6,  0;
%e A129533   0, 10, 30, 30, 10,  0;
%e A129533   0, 15, 60, 90, 60, 15, 0;
%e A129533   ...
%p A129533 dd:=proc(n,m) if m=0 or n=0 then 0 else (m+n)!/(2*(m-1)!*(n-1)!); fi; end;
%p A129533 f:=n->[seq(dd(n,m),m=0..30)];
%p A129533 for n from 0 to 10 do lprint(f(n)); od: # produces sequence as square array
%p A129533 T:=(n,k)->k*(k+1)*binomial(n,k+1)/2: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A129533 Table[Table[D[Expand[FunctionExpand[QBinomial[n, k, q]]], q] /. q -> 1, {k, 0, n}], {n, 0, 15}] // Grid (* _Geoffrey Critzer_, Mar 22 2018 *)
%Y A129533 Cf. A001788.
%Y A129533 Rows give A000004, A000217, A027480, A033487, A266732, A240440, A266733.
%Y A129533 A128503 and A094305 are very similar sequences.
%K A129533 nonn,tabl,changed
%O A129533 0,8
%A A129533 _Emeric Deutsch_, Apr 22 2007
%E A129533 Entry revised by _N. J. A. Sloane_, Jan 06 2016