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%I #17 Jun 29 2019 02:17:02
%S 1,0,1,0,4,3,0,24,40,15,0,192,520,420,105,0,1920,7392,9520,5040,945,0,
%T 23040,116928,211456,176400,69300,10395,0,322560,2055168,4858560,
%U 5642560,3465000,1081080,135135,0,5160960,39896064,117722880,177580480,150870720,73153080,18918900,2027025
%N Triangle read by rows, n>=0, k>=0, T(0,0) = 1, T(n,k) = Sum_{j=0..k} (C(n+k,k-j)*(-1)^(k-j)*2^(n-j)*Sum_{i=0..j} (C(n+j,i)*|S(n+j-i,j-i)|)), S = Stirling number of first kind.
%C This triangle was inspired by a formula of Vladimir Kruchinin given in A001662.
%F T(n,1) = A002866(n) for n>0.
%F T(n,n) = A001147(n).
%F Sum((-1)^(n-k)*T(n,k)) = A001662(n+1).
%e [n\k 0, 1, 2, 3, 4, 5]
%e [0] 1,
%e [1] 0, 1,
%e [2] 0, 4, 3,
%e [3] 0, 24, 40, 15,
%e [4] 0, 192, 520, 420, 105,
%e [5] 0, 1920, 7392, 9520, 5040, 945,
%p A201636 := proc(n,k) if n=0 and k=0 then 1 else
%p add(binomial(n+k,k-j)*(-1)^(n+k-j)*2^(n-j)*
%p add(binomial(n+j,i)*stirling1(n+j-i,j-i),i=0..j),j=0..k) fi end:
%p for n from 0 to 8 do print(seq(A201636(n,k),k=0..n)) od;
%t T[0, 0] = 1; T[n_, k_] := T[n, k] = Sum[Binomial[n+k, k-j]*(-1)^(n+k-j)* 2^(n-j)*Sum[Binomial[n+j, i]*StirlingS1[n+j-i, j-i], {i, 0, j}], {j, 0, k}];
%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Jean-François Alcover_, Jun 29 2019 *)
%o (Sage)
%o def A201636(n,k) :
%o if n==0 and k==0: return 1
%o return add(binomial(n+k,k-j)*(-1)^(k-j)*2^(n-j)*add(binomial(n+j,i)* stirling_number1(n+j-i,j-i) for i in (0..j)) for j in (0..k))
%Y Cf. A001662, A001147, A002866.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Nov 13 2012
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