cp's OEIS Frontend

A176157 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS2(n, n-j)*binomial(n, j), read by rows.

%I A176157 #25 Sep 08 2022 08:45:52
%S A176157 1,1,1,1,3,1,1,10,10,1,1,25,63,25,1,1,51,296,296,51,1,1,91,1060,2395,
%T A176157 1060,91,1,1,148,3081,14008,14008,3081,148,1,1,225,7665,62909,127883,
%U A176157 62909,7665,225,1,1,325,16948,230032,851758,851758,230032,16948,325,1
%N A176157 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS2(n, n-j)*binomial(n, j), read by rows.
%C A176157 Row sum are: {1, 2, 5, 22, 115, 696, 4699, 34476, 269483, 2198128, 18229726, ...}.
%C A176157 The first negative terms are T(14,6) = T(14,8) = -17062199622 = a(111), T(14,7) = -38263538781, T(15,5) = T(15,10) = -18803914339, T(15,6) = T(15,9) = -315758882649, T(15,7) = T(15,8) = -1027328563614. - _Georg Fischer_, _Hugo Pfoertner_, Jul 16 2020
%H A176157 G. C. Greubel, <a href="/A176157/b176157.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176157 With f(n, k) = Sum_{j=0..k} StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS2(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.
%e A176157 Triangle begins as:
%e A176157   1;
%e A176157   1,   1;
%e A176157   1,   3,     1;
%e A176157   1,  10,    10,      1;
%e A176157   1,  25,    63,     25,       1;
%e A176157   1,  51,   296,    296,      51,       1;
%e A176157   1,  91,  1060,   2395,    1060,      91,       1;
%e A176157   1, 148,  3081,  14008,   14008,    3081,     148,      1;
%e A176157   1, 225,  7665,  62909,  127883,   62909,    7665,    225,     1;
%e A176157   1, 325, 16948, 230032,  851758,  851758,  230032,  16948,   325,   1;
%e A176157   1, 451, 34191, 716796, 4390866, 7945116, 4390866, 716796, 34191, 451, 1;
%p A176157 with(combinat);
%p A176157 f:= proc(n, k) option remember; add(stirling2(n, n-j)*binomial(n, j), j=0..k) + add(stirling2(n, n-j)* binomial(n, j), j=0..n-k); end;
%p A176157 seq(seq(f(n,k) -f(n,0) +1, k=0..n), n=0..10); # _G. C. Greubel_, Nov 26 2019
%t A176157 f[n_, k_]:= Sum[StirlingS2[n, n-j]*Binomial[n, j], {j,0,k}] + Sum[StirlingS2[n, n-j]*Binomial[n, j], {j,0,n-k}];
%t A176157 Table[f[n, k] - f[n, 0] + 1, {n,0,10}, {k,0,n}]//Flatten
%o A176157 (PARI)
%o A176157 f(n,k) = sum(j=0,k, stirling(n,n-j,2)*binomial(n,j)) + sum(j=0,n-k, stirling(n, n-j,2)*binomial(n,j));
%o A176157 T(n,k) = f(n,k) - f(n,0) + 1; \\ _G. C. Greubel_, Nov 26 2019
%o A176157 (Magma)
%o A176157 f:= func< n,k | (&+[StirlingSecond(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[StirlingSecond(n,n-j)*Binomial(n,j): j in [0..n-k]]) >;
%o A176157 [f(n,k) - f(n,0) + 1: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 26 2019
%o A176157 (Sage)
%o A176157 def f(n, k): return sum(stirling_number2(n,n-j)*binomial(n,j) for j in (0..k)) + sum(stirling_number2(n, n-j)*binomial(n,j) for j in (0..n-k))
%o A176157 [[f(n, k)-f(n,0)+1 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 26 2019
%o A176157 (GAP)
%o A176157 f:= function(n,k) return Sum([0..k], j-> Stirling2(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> Stirling2(n, n-j)*Binomial(n,j)); end;
%o A176157 Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n,0)+1 ))); # _G. C. Greubel_, Nov 26 2019
%Y A176157 Cf. A008277, A176153, A176154, A176155, A176156.
%K A176157 sign,tabl
%O A176157 0,5
%A A176157 _Roger L. Bagula_, Apr 10 2010
%E A176157 Name edited by _G. C. Greubel_, Nov 26 2019