|
%I #6 Sep 08 2022 08:45:41
%S 4,1,1,1,4,1,1,15,15,1,1,76,249,76,1,1,485,3516,3516,485,1,1,3606,
%T 46623,101354,46623,3606,1,1,30247,617541,2388107,2388107,617541,
%U 30247,1,1,282248,8416315,51483931,91651662,51483931,8416315,282248,1,1,2903049,119667766,1071669632,3021085118,3021085118,1071669632,119667766,2903049,1
%N Triangle T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k), read by rows.
%H G. C. Greubel, <a href="/A155826/b155826.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k) * StirlingS1(n, n-k).
%F Sum_{k=0..n} T(n, k) = 2^n + 2*342111(n) + Sum_{k=0..n} binomial(k*(n-k), n). - _G. C. Greubel_, Jun 03 2021
%e Triangle begins as:
%e 4;
%e 1, 1;
%e 1, 4, 1;
%e 1, 15, 15, 1;
%e 1, 76, 249, 76, 1;
%e 1, 485, 3516, 3516, 485, 1;
%e 1, 3606, 46623, 101354, 46623, 3606, 1;
%e 1, 30247, 617541, 2388107, 2388107, 617541, 30247, 1;
%e 1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1;
%t T[n_, k_]:= Binomial[n, k] + Binomial[k*(n-k), n] + 2*(-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 03 2021 *)
%o (Magma)
%o A155826:= func< n,k | Binomial(n, k) + Binomial(k*(n-k), n) + 2*(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) >;
%o [A155826(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 03 2021
%o (Sage)
%o def A155826(n,k): return binomial(n, k) + binomial(k*(n-k), n) + 2*stirling_number1(n, k)*stirling_number1(n, n-k)
%o flatten([[A155826(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 03 2021
%Y Cf. A048994, A342111.
%K nonn,tabl
%O 0,1
%A _Roger L. Bagula_, Jan 28 2009
%E Edited by _G. C. Greubel_, Jun 03 2021
|
