A101492 - OEIS

%I #21 Sep 08 2022 08:45:16

%S 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 42,25,8,35,54,65,68,63,50,29,9,40,63,78,85,84,75,58,33,10,45,72,91,

%U 102,105,100,87,66,37,11,50,81,104,119,126,125,116,99,74,41,12,55,90,117

%N Triangle read by rows: T(n,k) = (n-k+1)*(4*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 5 0 0...

%C 1 5 9 0...

%C 1 5 9 13...

%C ...

%C T(n+0,0) = 1*n = A000027(n+1),

%C T(n+0,1) = 5*n = A008587(n),

%C T(n+1,2) = 9*n = A008591(n),

%C T(n+2,3) = 13*n = A008595(n),

%C so, for example,

%C T(n,n) = 4*n+1 = A016813(n),

%C T(n+1,n) = 8*n+2 = A017089(n),

%C T(n,0)*T(n,1)/10 = A000217(n) (triangular numbers),

%C T(n,n)*T(n,0) = A001107(n+1) (10-gonal numbers: 4*n^2 - 3*n),

%C T(n,n)*T(n,1)/5 = A007742(n).

%H Muniru A Asiru, <a href="/A101492/b101492.txt">Rows n=0..150 of triangle, flattened</a>

%t Flatten[Table[(n+1-k)(4k+1),{n,0,15},{k,0,n}]] (* _Harvey P. Dale_, Jun 09 2011 *)

%o (PARI) T(n, k) = if(k>n,0,(n-k+1)*(4*k+1));

%o for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

%o (GAP) Flat(List([0..11],n->List([0..n],k->(n+1-k)*(4*k+1)))); # _Muniru A Asiru_, Mar 07 2019

%o (Magma) [[(n+1-k)*(4*k+1): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Mar 07 2019

%o (Sage) [[(n-k+1)*(4*k+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 07 2019

%Y Row sums give hexagonal pyramidal numbers A002412.

%Y Cf. A101493 for product B*A, A002412.

%K nonn,tabl

%O 0,2

%A Lambert Klasen (lambert.klasen(AT)gmx.de) and _Gary W. Adamson_, Jan 21 2005