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A086329 Triangle T(n,k) read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] where DELTA is the operator defined in A084938.

%I A086329 #21 Jun 22 2022 02:52:49
%S A086329 1,0,1,0,1,1,0,1,4,1,0,1,9,11,1,0,1,16,48,26,1,0,1,25,140,202,57,1,0,
%T A086329 1,36,325,916,747,120,1,0,1,49,651,3045,5071,2559,247,1,0,1,64,1176,
%U A086329 8260,23480,25300,8362,502,1,0,1,81,1968,19404,84456,159736,117962,26520,1013,1
%N A086329 Triangle T(n,k) read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, ...] where DELTA is the operator defined in A084938.
%C A086329 See A087903 for another version (transposed). - _Philippe Deléham_, Jun 13 2004
%H A086329 G. C. Greubel, <a href="/A086329/b086329.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A086329 Sum_{k=0..n} T(n, k) = A086211(n, 0).
%F A086329 T(n, 1) = 1, n > 0.
%F A086329 T(n, 2) = (n-1)^2, n > 0.
%F A086329 T(k+1, k) = 2^(k+1) - k - 2 = A000295(k+1).
%F A086329 Sum_{k=0..n} T(n, k) = A074664(n+1). - _Philippe Deléham_, Jun 13 2004
%F A086329 Sum_{k=0..n} T(n,k)*2^k = A171151(n). - _Philippe Deléham_, Dec 05 2009
%F A086329 T(n, k) = A087903(n, n-k+1). - _G. C. Greubel_, Jun 21 2022
%e A086329 Triangle begins:
%e A086329   1;
%e A086329   0, 1;
%e A086329   0, 1,  1;
%e A086329   0, 1,  4,    1;
%e A086329   0, 1,  9,   11,    1;
%e A086329   0, 1, 16,   48,   26,     1;
%e A086329   0, 1, 25,  140,  202,    57,     1;
%e A086329   0, 1, 36,  325,  916,   747,   120,    1;
%e A086329   0, 1, 49,  651, 3045,  5071,  2559,  247,   1;
%e A086329   0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1; ...
%t A086329 T[n_, k_]:= T[n, k]= If[n==0, 1, StirlingS2[n, k] + Sum[(k-m-1)*T[n-j-1, k- m]*StirlingS2[j, m], {m,0,k-1}, {j,0,n-2}]];
%t A086329 A086329[n_, k_]:= T[n,n-k+1];
%t A086329 Table[A086329[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 21 2022 *)
%o A086329 (SageMath)
%o A086329 @CachedFunction
%o A086329 def T(n,k): # T=A087903
%o A086329     if (n==0): return 1
%o A086329     else: return stirling_number2(n, k) + sum( sum( (k-m-1)*T(n-j-1, k-m)*stirling_number2(j, m) for m in (0..k-1) ) for j in (0..n-2) )
%o A086329 def A086329(n,k): return T(n, n-k+1)
%o A086329 flatten([[A086329(n, k) for k in (0..n)] for n in (0..14)]) # _G. C. Greubel_, Jun 21 2022
%Y A086329 Cf. A000290, A000295, A074664, A084938, A086211, A171151.
%K A086329 easy,nonn,tabl
%O A086329 0,9
%A A086329 _Philippe Deléham_, Aug 30 2003, Jun 12 2007