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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.
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%I #21 Jun 22 2022 02:52:49

%S 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 1,36,325,916,747,120,1,0,1,49,651,3045,5071,2559,247,1,0,1,64,1176,

%U 8260,23480,25300,8362,502,1,0,1,81,1968,19404,84456,159736,117962,26520,1013,1

%N 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 See A087903 for another version (transposed). - _Philippe Deléham_, Jun 13 2004

%H G. C. Greubel, <a href="/A086329/b086329.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Sum_{k=0..n} T(n, k) = A086211(n, 0).

%F T(n, 1) = 1, n > 0.

%F T(n, 2) = (n-1)^2, n > 0.

%F T(k+1, k) = 2^(k+1) - k - 2 = A000295(k+1).

%F Sum_{k=0..n} T(n, k) = A074664(n+1). - _Philippe Deléham_, Jun 13 2004

%F Sum_{k=0..n} T(n,k)*2^k = A171151(n). - _Philippe Deléham_, Dec 05 2009

%F T(n, k) = A087903(n, n-k+1). - _G. C. Greubel_, Jun 21 2022

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 1, 4, 1;

%e 0, 1, 9, 11, 1;

%e 0, 1, 16, 48, 26, 1;

%e 0, 1, 25, 140, 202, 57, 1;

%e 0, 1, 36, 325, 916, 747, 120, 1;

%e 0, 1, 49, 651, 3045, 5071, 2559, 247, 1;

%e 0, 1, 64, 1176, 8260, 23480, 25300, 8362, 502, 1; ...

%t 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[n_, k_]:= T[n,n-k+1];

%t Table[A086329[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 21 2022 *)

%o (SageMath)

%o @CachedFunction

%o def T(n,k): # T=A087903

%o if (n==0): return 1

%o 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 def A086329(n,k): return T(n, n-k+1)

%o flatten([[A086329(n, k) for k in (0..n)] for n in (0..14)]) # _G. C. Greubel_, Jun 21 2022

%Y Cf. A000290, A000295, A074664, A084938, A086211, A171151.

%K easy,nonn,tabl

%O 0,9

%A _Philippe Deléham_, Aug 30 2003, Jun 12 2007