A323956 - OEIS

%I #20 Sep 08 2022 08:46:23

%S 1,3,1,7,4,1,13,9,5,1,21,16,11,6,1,31,25,19,13,7,1,43,36,29,22,15,8,1,

%T 57,49,41,33,25,17,9,1,73,64,55,46,37,28,19,10,1,91,81,71,61,51,41,31,

%U 21,11,1,111,100,89,78,67,56,45,34,23,12,1

%N Triangle read by rows: T(n, k) = 1 + n * (n - k) for 1 <= k <= n.

%H G. C. Greubel, <a href="/A323956/b323956.txt">Rows n = 1..100 of triangle, flattened</a>

%F From _Werner Schulte_, Feb 12 2019: (Start)

%F G.f.: Sum_{n>0,k=1..n} T(n,k)*x^k*t^n = x*t*((1-t+2*t^2)*(1-x*t) + (1-t)*t)/((1-t)^3*(1-x*t)^2).

%F Row sums: Sum_{k=1..n} T(n,k) = A006000(n-1) for n > 0.

%F Recurrence: T(n,k) = T(n,k-1) - n for 1 < k <= n with initial values T(n,1) = n^2-n+1 for n > 0.

%F Recurrence: T(n,k) = T(n-1,k) + 2*n-k-1 for 1 <= k < n with initial values T(n,n) = 1 for n > 0.

%F (End)

%e Triangle begins:

%e   n\k:   1   2   3   4   5   6   7   8   9  10  11  12

%e   ====================================================

%e     1:   1

%e     2:   3   1

%e     3:   7   4   1

%e     4:  13   9   5   1

%e     5:  21  16  11   6   1

%e     6:  31  25  19  13   7   1

%e     7:  43  36  29  22  15   8   1

%e     8:  57  49  41  33  25  17   9   1

%e     9:  73  64  55  46  37  28  19  10   1

%e    10:  91  81  71  61  51  41  31  21  11   1

%e    11: 111 100  89  78  67  56  45  34  23  12   1

%e    12: 133 121 109  97  85  73  61  49  37  25  13   1

%e   etc.

%t Table[1+n*(n-k),{n,12},{k,n}]//Flatten

%o (PARI) {T(n,k) = 1+n*(n-k)}; \\ _G. C. Greubel_, Apr 22 2019

%o (Magma) [[1+n*(n-k): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Apr 22 2019

%o (Sage) [[1+n*(n-k) for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Apr 22 2019

%Y First column is A002061. Second column is A000290. Third column is A028387.

%Y Cf. A000126, A001610, A001644, A006000, A169985, A306351, A323952, A323955.

%K nonn,tabl

%O 1,2

%A _Gus Wiseman_, Feb 10 2019