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A101492 Triangle read by rows: T(n,k) = (n-k+1)*(4*k+1).

%I A101492 #21 Sep 08 2022 08:45:16
%S A101492 1,2,5,3,10,9,4,15,18,13,5,20,27,26,17,6,25,36,39,34,21,7,30,45,52,51,
%T A101492 42,25,8,35,54,65,68,63,50,29,9,40,63,78,85,84,75,58,33,10,45,72,91,
%U A101492 102,105,100,87,66,37,11,50,81,104,119,126,125,116,99,74,41,12,55,90,117
%N A101492 Triangle read by rows: T(n,k) = (n-k+1)*(4*k+1).
%C A101492 The triangle is generated from the product A*B
%C A101492 of the infinite lower triangular matrices A =
%C A101492 1 0 0 0...
%C A101492 1 1 0 0...
%C A101492 1 1 1 0...
%C A101492 1 1 1 1...
%C A101492 ... and B =
%C A101492 1 0 0 0...
%C A101492 1 5 0 0...
%C A101492 1 5 9 0...
%C A101492 1 5 9 13...
%C A101492 ...
%C A101492 T(n+0,0) = 1*n = A000027(n+1),
%C A101492 T(n+0,1) = 5*n = A008587(n),
%C A101492 T(n+1,2) = 9*n = A008591(n),
%C A101492 T(n+2,3) = 13*n = A008595(n),
%C A101492 so, for example,
%C A101492 T(n,n) = 4*n+1 = A016813(n),
%C A101492 T(n+1,n) = 8*n+2 = A017089(n),
%C A101492 T(n,0)*T(n,1)/10 = A000217(n) (triangular numbers),
%C A101492 T(n,n)*T(n,0) = A001107(n+1) (10-gonal numbers: 4*n^2 - 3*n),
%C A101492 T(n,n)*T(n,1)/5 = A007742(n).
%H A101492 Muniru A Asiru, <a href="/A101492/b101492.txt">Rows n=0..150 of triangle, flattened</a>
%t A101492 Flatten[Table[(n+1-k)(4k+1),{n,0,15},{k,0,n}]] (* _Harvey P. Dale_, Jun 09 2011 *)
%o A101492 (PARI) T(n, k) = if(k>n,0,(n-k+1)*(4*k+1));
%o A101492 for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())
%o A101492 (GAP) Flat(List([0..11],n->List([0..n],k->(n+1-k)*(4*k+1)))); # _Muniru A Asiru_, Mar 07 2019
%o A101492 (Magma) [[(n+1-k)*(4*k+1): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Mar 07 2019
%o A101492 (Sage) [[(n-k+1)*(4*k+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 07 2019
%Y A101492 Row sums give hexagonal pyramidal numbers A002412.
%Y A101492 Cf. A101493 for product B*A, A002412.
%K A101492 nonn,tabl
%O A101492 0,2
%A A101492 Lambert Klasen (lambert.klasen(AT)gmx.de) and _Gary W. Adamson_, Jan 21 2005