A154843 - OEIS

%I #12 Sep 08 2022 08:45:40

%S 2,1,1,1,-2,1,1,-1,-1,1,1,-12,22,-12,1,1,14,-15,-15,14,1,1,-135,359,

%T -450,359,-135,1,1,699,-1589,889,889,-1589,699,1,1,-5068,13390,-15092,

%U 13538,-15092,13390,-5068,1,1,40284,-109038,113588,-44835,-44835,113588,-109038,40284,1

%N Triangular array, T(n,k) = s(n,k) + s(n,n-k), where s(n,k) are the Stirling numbers of the first kind.

%C Except for the first two rows the row sums are zero.

%H G. C. Greubel, <a href="/A154843/b154843.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = s(n, k) + s(n, n - k), where s(n,k) are the Stirling numbers of the first kind (A048994).

%e Triangle begins as:

%e   2;

%e   1,     1;

%e   1,    -2,     1;

%e   1,    -1,    -1,      1;

%e   1,   -12,    22,    -12,     1;

%e   1,    14,   -15,    -15,    14,      1;

%e   1,  -135,   359,   -450,   359,   -135,     1;

%e   1,   699, -1589,    889,   889,  -1589,   699,     1;

%e   1, -5068, 13390, -15092, 13538, -15092, 13390, -5068, 1;

%t Table[StirlingS1[n, k] +StirlingS1[n, n-k], {n,0,10}, {k,0,n} ]//Flatten (* modified by _G. C. Greubel_, Apr 07 2019 *)

%o (PARI) {T(n,k) = stirling(n,k,1) + stirling(n,n-k,1)}; \\ _G. C. Greubel_, Apr 07 2019

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

%o (Sage) [[(-1)^(n-k)*(stirling_number1(n,k) + (-1)^n*stirling_number1(n,n-k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 07 2019

%Y Cf. A048994.

%K tabl,sign

%O 0,1

%A _Roger L. Bagula_, Jan 16 2009