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%I #19 Sep 08 2022 08:45:41
%S 1,1,2,1,3,19,1,4,41,274,1,5,26,812,5521,1,6,-370,1000,20490,143828,1,
%T 7,-3023,-8607,34062,640356,4607296,1,8,-16977,-97974,-192901,1249164,
%U 23929389,175377146,1,9,-83108,-703130,-1227484,-8076692,53594570,1040938950,7739288201
%N Triangle read by rows: t(n,m) = Sum_{i=0..n} (-1)^(m-i)*Eulerian1(n-i+1, m-i) *Stirling2(n+i+1, i+1), where Eulerian1 are the Eulerian numbers of the first kind (A173018).
%C Row sums are: {1, 3, 23, 320, 6365, 164955, 5270092, 200247856, 8823731317, 442515406465, 24893699411935,...}.
%C This sequence results from a substitution of the Eulerian numbers for the binomial in Smiley's Corollary 5.
%H G. C. Greubel, <a href="/A156364/b156364.txt">Rows n=0..100 of triangle, flattened</a>
%H L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/BSfront.html">Completion of a Rational Function Sequence of Carlitz</a>, page 3.
%F t(n,m) = Sum_{i=0..n} (-1)^(m-i)*Eulerian1(n-i+1, m-i)*Stirling2(n+i+1, i+1), where Eulerian1(n,k) = Sum_{j=0..k+1} (-1)^j * binomial(n+1, j)*(k+1-j)^n or Eulerian1(n,k) = A173018(n,k).
%e Triangle begins as:
%e 1;
%e 1, 2;
%e 1, 3, 19;
%e 1, 4, 41, 274;
%e 1, 5, 26, 812, 5521;
%e 1, 6, -370, 1000, 20490, 143828;
%e 1, 7, -3023, -8607, 34062, 640356, 4607296;
%e 1, 8, -16977, -97974, -192901, 1249164, 23929389, 175377146;
%t Eulerian1[n_, k_]:= Sum[(-1)^j Binomial[n+1, j](k+1-j)^n, {j, 0, k+1}];
%t t[n_, m_]:= Sum[(-1)^(m-i)*Eulerian1[n-i+1,m-i]*StirlingS2[n+i+1,i+1], {i,0,n}];
%t Table[t[n, m], {n,0,10}, {m,0,n}]//Flatten
%o (PARI)
%o Eulerian1(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1, j)*(k+1-j)^n);
%o t(n,m) = sum(i=0,n, (-1)^(m-i)*Eulerian1(n-i+1,m-i)*stirling(n+i+1, i+1,2));
%o for(n=0,10, for(m=0,n, print1(t(n,m), ", "))) \\ _G. C. Greubel_, Feb 24 2019
%o (Magma) [[(&+[(-1)^(m-j)*StirlingSecond(n+j+1, j+1)*(&+[(-1)^k*Binomial(n-j+2,k)*(m-j+1-k)^(n-j+1): k in [0..m-j+1]]): j in [0..m]]): m in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Feb 25 2019
%o (Sage) [[sum((-1)^(m-j)*stirling_number2(n+j+1, j+1)*sum( (-1)^k*binomial(n-j+2, k)*(m-j-k+1)^(n-j+1) for k in (0..m-j+1)) for j in (0..m)) for m in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 25 2019
%Y Cf. A048993 (Stirling2), A156139, A156363, A173018.
%K sign,tabl
%O 0,3
%A _Roger L. Bagula_, Feb 08 2009
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