cp's OEIS Frontend

A176156 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^j*StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} (-1)^j*StirlingS2(n, n-j)*binomial(n, j), read by rows.

%I A176156 #18 Sep 08 2022 08:45:52
%S A176156 1,1,1,1,3,1,1,10,10,1,1,25,67,25,1,1,51,281,281,51,1,1,91,646,1036,
%T A176156 646,91,1,1,148,-1217,-12536,-12536,-1217,148,1,1,225,-31079,-287223,
%U A176156 -548785,-287223,-31079,225,1,1,325,-342899,-3906899,-11000741,-11000741,-3906899,-342899,325,1
%N A176156 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^j*StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} (-1)^j*StirlingS2(n, n-j)*binomial(n, j), read by rows.
%C A176156 Row sum are: {1, 2, 5, 22, 119, 666, 2512, -27208, -1184937, -30500426, -716845999, ...}.
%H A176156 G. C. Greubel, <a href="/A176156/b176156.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176156 With f(n, k) = Sum_{j=0..k} (-1)^j*StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} (-1)^j*StirlingS1(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.
%e A176156 Triangle begins as:
%e A176156   1;
%e A176156   1,   1;
%e A176156   1,   3,      1;
%e A176156   1,  10,     10,       1;
%e A176156   1,  25,     67,      25,       1;
%e A176156   1,  51,    281,     281,      51,       1;
%e A176156   1,  91,    646,    1036,     646,      91,      1;
%e A176156   1, 148,  -1217,  -12536,  -12536,   -1217,    148,   1;
%e A176156   1, 225, -31079, -287223, -548785, -287223, -31079, 225, 1;
%p A176156 with(combinat);
%p A176156 f:= proc(n, k) option remember; add((-1)^j*stirling1(n, n-j)*binomial(n, j), j=0..k) + add((-1)^j*stirling1(n, n-j)* binomial(n, j), j=0..n-k); end;
%p A176156 seq(seq(f(n,k) -f(n,0) +1, k=0..n), n=0..10); # _G. C. Greubel_, Nov 26 2019
%t A176156 f[n_, k_]:= Sum[(-1)^j*StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}] + Sum[(-1)^j*StirlingS1[n, n-j]*Binomial[n, j], {j,0,n-k}];
%t A176156 Table[f[n, k] - f[n, 0] + 1, {n,0,10}, {k,0,n}]//Flatten
%o A176156 (PARI)
%o A176156 f(n,k) = sum(j=0,k, (-1)^j*stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, (-1)^j*stirling(n, n-j,1)*binomial(n,j));
%o A176156 T(n,k) = f(n,k) - f(n,0) + 1; \\ _G. C. Greubel_, Nov 26 2019
%o A176156 (Magma)
%o A176156 f:= func< n,k | (&+[(-1)^j*StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[(-1)^j*StirlingFirst(n,n-j)*Binomial(n,j): j in [0..n-k]]) >;
%o A176156 [f(n,k) - f(n,0) + 1: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 26 2019
%o A176156 (Sage)
%o A176156 def f(n, k): return sum(stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum(stirling_number1(n, n-j)*binomial(n,j) for j in (0..n-k))
%o A176156 [[f(n, k)-f(n,0)+1 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 26 2019
%o A176156 (GAP)
%o A176156 f:= function(n,k) return Sum([0..k], j-> Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> Stirling1(n, n-j)*Binomial(n,j)); end;
%o A176156 Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n,0)+1 ))); # _G. C. Greubel_, Nov 26 2019
%Y A176156 Cf. A048994, A132393, A176153, A176154, A176155, A176157.
%K A176156 sign,tabl
%O A176156 0,5
%A A176156 _Roger L. Bagula_, Apr 10 2010
%E A176156 Name edited by _G. C. Greubel_, Nov 26 2019