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%I #12 Sep 08 2022 08:45:52
%S 1,1,1,1,-1,1,1,-8,-8,1,1,-23,67,-23,1,1,-49,181,181,-49,1,1,-89,1906,
%T -6704,1906,-89,1,1,-146,-1511,9808,9808,-1511,-146,1,1,-223,49113,
%U -426551,782671,-426551,49113,-223,1,1,-323,-343547,3220453,-3873389,-3873389,3220453,-343547,-323,1
%N Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j), read by rows.
%C Row sum are: {1, 2, 1, -14, 23, 266, -3068, 16304, 27351, -1993610, 31213301, ...}.
%H G. C. Greubel, <a href="/A176155/b176155.txt">Rows n = 0..100 of triangle, flattened</a>
%F With f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -1, 1;
%e 1, -8, -8, 1;
%e 1, -23, 67, -23, 1;
%e 1, -49, 181, 181, -49, 1;
%e 1, -89, 1906, -6704, 1906, -89, 1;
%e 1, -146, -1511, 9808, 9808, -1511, -146, 1;
%e 1, -223, 49113, -426551, 782671, -426551, 49113, -223, 1;
%e 1, -323, -343547, 3220453, -3873389, -3873389, 3220453, -343547, -323, 1;
%p with(combinat);
%p f:= proc(n, k) option remember; add(stirling1(n, n-j)*binomial(n, j), j=0..k) + add(stirling1(n, n-j)* binomial(n, j), j=0..n-k); end;
%p seq(seq(f(n,k) -f(n,0) +1, k=0..n), n=0..10); # _G. C. Greubel_, Nov 26 2019
%t f[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}] + Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,n-k}];
%t Table[f[n, k] - f[n, 0] + 1, {n,0,10}, {k,0,n}]//Flatten
%o (PARI)
%o f(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, stirling(n, n-j,1)*binomial(n,j));
%o T(n,k) = f(n,k) - f(n,0) + 1; \\ _G. C. Greubel_, Nov 26 2019
%o (Magma)
%o f:= func< n,k | (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..n-k]]) >;
%o [f(n,k) - f(n,0) + 1: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 26 2019
%o (Sage)
%o def f(n, k): return sum((-1)^j*stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum((-1)^j*stirling_number1(n, n-j)*binomial(n,j) for j in (0..n-k))
%o [[f(n, k)-f(n,0)+1 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 26 2019
%o (GAP)
%o f:= function(n,k) return Sum([0..k], j-> (-1)^j*Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> (-1)^j*Stirling1(n, n-j)*Binomial(n,j)); end;
%o Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n,0)+1 ))); # _G. C. Greubel_, Nov 26 2019
%Y Cf. A048994, A132393, A176153, A176154, A176156, A176157.
%K sign,tabl
%O 0,8
%A _Roger L. Bagula_, Apr 10 2010
%E Name edited by _G. C. Greubel_, Nov 27 2019
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