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 A122101 T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A000670(n-k+i). 3

%I

%S 1,1,0,3,2,2,13,10,8,6,75,62,52,44,38,541,466,404,352,308,270,4683,

%T 4142,3676,3272,2920,2612,2342,47293,42610,38468,34792,31520,28600,

%U 25988,23646,545835,498542,455932,417464,382672,351152,322552,296564,272918

%N T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A000670(n-k+i).

%H G. C. Greubel, <a href="/A122101/b122101.txt">Rows n = 0..100 of triangle, flattened</a>

%F Doubly-exponential generating function: Sum_{n, k} a(n-k,k) x^n/n! y^k/k! = exp(-y)/(2-exp(x+y)).

%e Triangle begins as:

%e 1;

%e 1, 0;

%e 3, 2, 2;

%e 13, 10, 8, 6;

%e 75, 62, 52, 44, 38;

%e 541, 466, 404, 352, 308, 270;

%e 4683, 4142, 3676, 3272, 2920, 2612, 2342;

%e ...

%p T:= (n, k)-> k!*(n-k)!*coeff(series(coeff(series(exp(-y)/

%p (2-exp(x+y)), y, k+1), y, k), x, n-k+1), x, n-k):

%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Oct 02 2019

%p # second Maple program:

%p b:= proc(n) option remember; `if`(n<2, 1,

%p end:

%p T:= (n, k)-> add(binomial(k, j)*(-1)^j*b(n-j), j=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..12); # _Alois P. Heinz_, Oct 02 2019

%t A000670[n_]:= If[n==0,1,Sum[k! StirlingS2[n, k], {k, n}]]; T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[k, j]*A000670[n-k+j], {j,0,k}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 02 2019 *)

%o (PARI)

%o A000670(n) = sum(k=0,n, k!*stirling(n,k,2));

%o T(n,k) = sum(j=0,k, (-1)^(k-j)*binomial(k, j)*A000670(n-k+j));

%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 02 2019

%o (MAGMA)

%o A000670:= func< n | &+[Factorial(k)*StirlingSecond(n,k): k in [0..n]] >;

%o [(&+[(-1)^(k-j)*Binomial(k,j)*A000670(n-k+j): j in [0..k]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Oct 02 2019

%o (Sage)

%o def A000670(n): return sum(factorial(k)*stirling_number2(n,k) for k in (0..n))

%o def T(n,k): return sum((-1)^(k-j)*binomial(k, j)*A000670(n-k+j) for j in (0..k))

%o [[T(n,k) for k in (0..n)] for n in (0..10)]

%o (GAP)

%o A000670:= function(n)

%o return Sum([0..n], i-> Factorial(i)*Stirling2(n,i) ); end;

%o T:= function(n,k)

%o return Sum([0..k], j-> (-1)^(k-j)*Binomial(k, j)*A000670(n-k+j) ); end;

%o Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Oct 02 2019

%Y Columns k=0-1 give: A000670, A232472.

%Y Row sums give A089677(n+1).

%Y Main diagonal gives A052841.

%Y Cf. A005649, A069321, A073146.

%K easy,nonn,tabl

%O 0,4

%A _Vladeta Jovovic_, Oct 18 2006

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Last modified February 17 23:35 EST 2020. Contains 332006 sequences. (Running on oeis4.)