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%I #8 Mar 14 2015 17:05:29
%S 1,2,4,3,8,7,4,12,14,10,5,16,21,20,13,6,20,28,30,26,16,7,24,35,40,39,
%T 32,19,8,28,42,50,52,48,38,22,9,32,49,60,65,64,57,44,25,10,36,56,70,
%U 78,80,76,66,50,28,11,40,63,80,91,96,95,88,75,56,31,12,44,70,90,104,112,114
%N Triangle read by rows: T(n,k)=(n+1-k)*(3*k+1).
%C The triangle is generated from the product A*B
%C of the infinite lower triangular matrices A =
%C 1 0 0 0...
%C 1 1 0 0...
%C 1 1 1 0...
%C 1 1 1 1...
%C ... and B =
%C 1 0 0 0...
%C 1 4 0 0...
%C 1 4 7 0...
%C 1 4 7 10...
%C ...
%C Row sums give pentagonal pyramidal numbers A002411 T(n+0,0)= 1*n=A000027(n) T(n+0,1)= 4*n=A008586(n) T(n+1,2)= 7*n=A008589(n) T(n+2,3)=10*n=A008592(n) ...
%C so for example T(n+1,n-0)=6*n+2=A016933(n) T(n+1,n-1)=9*n+3=A017197(n) T(n+2,n-1)=12*n+4=A017569(n)
%C T(n,0)*T(n,1) = A033996(n) (8 times triangular numbers)
%C T(n,n)*T(n,0) = A000567(n+1) (Octagonal numbers) etc.
%e Triangle begins:
%e 1,
%e 2, 4,
%e 3, 8, 7,
%e 4, 12, 14, 10,
%e 5, 16, 21, 20, 13,
%e 6, 20, 28, 30, 26, 16,
%e 7, 24, 35, 40, 39, 32, 19,
%e 8, 28, 42, 50, 52, 48, 38, 22,
%e 9, 32, 49, 60, 65, 64, 57, 44, 25,
%e 10, 36, 56, 70, 78, 80, 76, 66, 50, 28,
%e 11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, etc.
%e [_Bruno Berselli_, Feb 10 2014]
%t t[n_, k_] := If[n < k, 0, (3*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* _Robert G. Wilson v_, Jan 21 2005 *)
%o (PARI) T(n,k)=if(k>n,0,(n-k+1)*(3*k+1)) for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())
%Y Cf. A095871 (product B*A), A002411.
%K nonn,tabl
%O 0,2
%A Lambert Klasen (lambert.klasen(AT)gmx.de) and _Gary W. Adamson_, Jan 21 2005
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