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%I #19 Sep 08 2022 08:45:54
%S 2,4,-3,8,-5,4,16,1,11,-5,32,51,46,-14,6,64,281,337,22,22,-7,128,1163,
%T 2472,1121,176,-27,8,256,4257,15703,15493,4419,163,37,-9,512,14563,
%U 88354,155980,88486,14398,622,-44,10,1024,47785,455357,1310024,1310816,454730,48170,848,56,-11
%N Triangle T(n,k) read by rows: T(n,k) = Sum_{j=0..k} (-1)^j *binomial(n+1,j)*(k+2-j)^n, 0 <= k < n.
%C Row sums are apparently the 2nd column of A156984, 2, 1, 7, 23,...
%C Generalizes A008292 in the sense that a term "2" is added to the factor that is raised to the n-th power in the sum of the definition. A term "1" would generate A008292, too (up to index shifts).
%D B. Harris and C J. Park, A generalization of Eulerian numbers with a probabilistic Application, Statistics and Probability Letters 20 (1994), page 40
%H G. C. Greubel, <a href="/A180246/b180246.txt">Rows n=1..100 of triangle, flattened</a>
%F T(n,k) = Sum_{j=0..k} (-1)^j *binomial(n+1,j)*(k+2-j)^n, for k=0..n-1.
%e Triangle begins with:
%e 2;
%e 4, -3;
%e 8, -5, 4;
%e 16, 1, 11, -5;
%e 32, 51, 46, -14, 6;
%e 64, 281, 337, 22, 22, -7;
%e 128, 1163, 2472, 1121, 176, -27, 8;
%e 256, 4257, 15703, 15493, 4419, 163, 37, -9;
%e 512, 14563, 88354, 155980, 88486, 14398, 622, -44, 10;
%e 1024, 47785, 455357, 1310024, 1310816, 454730, 48170, 848, 56, -11;
%e ...
%p A180246 := proc(n,k) add( (-1)^v*binomial(n+1,v)*(k+2-v)^n,v=0..k) ; end proc: # _R. J. Mathar_, Jan 29 2011
%p P := proc(n,x) option remember; if n = 0 then 1 else
%p (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
%p expand(%) fi end:
%p A180246 := (n,k) -> coeff(P(n,x),x,k):
%p seq(print(seq(A180246(n,k),k=0..n-1)),n=0..10); # _Peter Luschny_, Mar 07 2014
%t t[n_, j_, d_]:= Sum[(-1)^v *Binomial[n+1, v](j+d-v)^n, {v, 0, j}];
%t Table[Flatten[Table[Table[t[n,k,m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
%t (* This sequence corresponds to m=2 *)
%t Table[Sum[(-1)^j*Binomial[n+1, j]*(k-j+2)^n, {j,0,k}], {n,1,12}, {k,0,n-1}]//Flatten
%o (PARI) {T(n,k) = sum(j=0,k, (-1)^j*binomial(n+1, j)*(k-j+2)^n)};
%o for(n=1,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 23 2019
%o (Magma) [[(&+[(-1)^j*Binomial(n+1, j)*(k-j+2)^n: j in [0..k]]): k in [0..n-1]]: n in [1..12]]; // _G. C. Greubel_, Feb 23 2019
%o (Sage) [[sum((-1)^j*binomial(n+1, j)*(k-j+2)^n for j in (0..k)) for k in (0..n-1)] for n in (1..12)] # _G. C. Greubel_, Feb 23 2019
%o (GAP) Flat(List([1..12], n-> List([0..n-1], k-> Sum([0..k], j-> (-1)^j*Binomial(n+1, j)*(k-j+2)^n )))); # _G. C. Greubel_, Feb 23 2019
%Y Cf. A008292, A157011.
%K sign,tabl,easy
%O 1,1
%A _Roger L. Bagula_, Aug 19 2010
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