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A158823 Triangle read by rows: matrix product A004736 * A158821.

%I A158823 #16 Apr 01 2021 14:37:54
%S A158823 1,3,1,6,2,2,10,3,4,3,15,4,6,6,4,21,5,8,9,8,5,28,6,10,12,12,10,6,36,7,
%T A158823 12,15,16,15,12,7,45,8,14,18,20,20,18,14,8,55,9,16,21,24,25,24,21,16,
%U A158823 9,66,10,18,24,28,30,30,28,24,18,10,78,11,20,27,32,35,36,35,32,27,20,11
%N A158823 Triangle read by rows: matrix product A004736 * A158821.
%H A158823 G. C. Greubel, <a href="/A158823/b158823.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A158823 Sum_{k=1..n} T(n, k) = A000292(n).
%F A158823 T(n, k) = Sum_{j=k..n} A004736(n, j)*A158821(j-1, k-1).
%F A158823 From _R. J. Mathar_, Mar 03 2011: (Start)
%F A158823 T(n, k) = (n-k+1)*(k-1), k>1.
%F A158823 T(n, 1) = A000217(n). (End)
%e A158823 First few rows of the triangle =
%e A158823    1;
%e A158823    3,  1;
%e A158823    6,  2,  2;
%e A158823   10,  3,  4,  3;
%e A158823   15,  4,  6,  6,  4;
%e A158823   21,  5,  8,  9,  8,  5;
%e A158823   28,  6, 10, 12, 12, 10,  6;
%e A158823   36,  7, 12, 15, 16, 15, 12,  7;
%e A158823   45,  8, 14, 18, 20, 20, 18, 14,  8;
%e A158823   55,  9, 16, 21, 24, 25, 24, 21, 16,  9;
%e A158823   66, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10;
%e A158823   78, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11;
%e A158823   91, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12;
%p A158823 A158823 := proc(n,m) add( A004736(n,k)*A158821(k-1,m-1),k=1..n) ; end: seq(seq(A158823(n,m),m=1..n),n=1..8) ; # _R. J. Mathar_, Oct 22 2009
%t A158823 Table[If[k==1, Binomial[n+1, 2], (n-k+1)*(k-1)], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Apr 01 2021 *)
%o A158823 (Magma) [k eq 1 select Binomial(n+1, 2) else (n-k+1)*(k-1): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Apr 01 2021
%o A158823 (Sage) flatten([[binomial(n+1, 2) if k==1 else (n-k+1)*(k-1) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Apr 01 2021
%Y A158823 Cf. A000292 (row sums), A003991, A004736, A158821.
%K A158823 nonn,tabl,easy
%O A158823 1,2
%A A158823 _Gary W. Adamson_ & _Roger L. Bagula_, Mar 28 2009
%E A158823 Corrected A-number in a formula - _R. J. Mathar_, Oct 30 2009