cp's OEIS Frontend

A176155 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.

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