cp's OEIS Frontend

A124932 Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).

%I A124932 #15 Sep 08 2022 08:45:28
%S A124932 1,2,3,3,9,6,4,18,24,10,5,30,60,50,15,6,45,120,150,90,21,7,63,210,350,
%T A124932 315,147,28,8,84,336,700,840,588,224,36,9,108,504,1260,1890,1764,1008,
%U A124932 324,45,10,135,720,2100,3780,4410,3360,1620,450,55
%N A124932 Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).
%C A124932 Row sums = A001793: (1, 5, 18, 56, 160, 432, ...).
%C A124932 Triangle is P*M, where P is the Pascal triangle as an infinite lower triangular matrix and M is an infinite bidiagonal matrix with (1,3,6,10,...) in the main diagonal and in the subdiagonal.
%C A124932 This number triangle can be used as a control sequence when listing combinations of subsets as in Pascals triangle by assigning a number to each element that corresponds to the n:th subset that the element belongs to. One then gets number blocks whose sums are the terms in this number triangle. - _Mats Granvik_, Jan 14 2009
%H A124932 G. C. Greubel, <a href="/A124932/b124932.txt">Rows n = 1..100 of triangle, flattened</a>
%F A124932 T(n,k) = binomial(k+1,2)*binomial(n,k). - _G. C. Greubel_, Nov 19 2019
%e A124932 First few rows of the triangle:
%e A124932   1;
%e A124932   2,   3;
%e A124932   3,   9,   6;
%e A124932   4,  18,  24,  10;
%e A124932   5,  30,  60,  50,  15;
%e A124932   6,  45, 120, 150,  90,  21;
%e A124932   7,  63, 210, 350, 315, 147,  28;
%e A124932   ...
%e A124932 From _Mats Granvik_, Dec 18 2009: (Start)
%e A124932 The numbers in this triangle are sums of the following recursive number blocks:
%e A124932 1................................
%e A124932 .................................
%e A124932 11.....12........................
%e A124932 .................................
%e A124932 111....112....123................
%e A124932 .......122.......................
%e A124932 .................................
%e A124932 1111...1112...1123...1234........
%e A124932 .......1122...1223...............
%e A124932 .......1222...1233...............
%e A124932 .................................
%e A124932 11111..11112..11123..11234..12345
%e A124932 .......11122..11223..12234.......
%e A124932 .......11222..12223..12334.......
%e A124932 .......12222..11233..12344.......
%e A124932 ..............12233..............
%e A124932 ..............12333..............
%e A124932 .................................
%e A124932 (End)
%p A124932 T:=(n,k)->k*(k+1)*binomial(n,k)/2: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t A124932 Table[Binomial[k + 1, 2]*Binomial[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Nov 19 2019 *)
%o A124932 (PARI) T(n,k) = binomial(k+1,2)*binomial(n,k); \\ _G. C. Greubel_, Nov 19 2019
%o A124932 (Magma) B:=Binomial; [B(k+1,2)*B(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Nov 19 2019
%o A124932 (Sage) b=binomial; [[b(k+1,2)*b(n,k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Nov 19 2019
%o A124932 (GAP) B:=Binomial;; Flat(List([1..12], n-> List([1..n], k-> B(k+1,2)* B(n,k) ))); # _G. C. Greubel_, Nov 19 2019
%Y A124932 Cf. A001793.
%K A124932 nonn,tabl
%O A124932 1,2
%A A124932 _Gary W. Adamson_, Nov 12 2006
%E A124932 Edited by _N. J. A. Sloane_, Nov 24 2006