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A211235 Array of generalized Eulerian numbers C(n,k) read by antidiagonals.

%I A211235 #31 Feb 18 2025 21:18:57
%S A211235 1,1,2,1,4,3,1,7,10,4,1,12,27,20,5,1,21,69,77,35,6,1,38,176,272,182,
%T A211235 56,7,1,71,456,936,846,378,84,8,1,136,1205,3210,3750,2232,714,120,9,1,
%U A211235 265,3247,11075,16290,12342,5214,1254,165,10
%N A211235 Array of generalized Eulerian numbers C(n,k) read by antidiagonals.
%H A211235 D. H. Lehmer, <a href="https://doi.org/10.1016/0097-3165(82)90020-6">Generalized Eulerian numbers</a>, J. Combin. Theory Ser.A 32 (1982), no. 2, 195-215. MR0654621 (83k:10026).
%F A211235 From _Peter Bala_, Oct 27 2015: (Start)
%F A211235 O.g.f. of n-th row of square array: 1/(1 - x)^n * (x*d/dx)^n log(1/(1 - x)), for n >= 1.
%F A211235 E.g.f. of square array: log((1 - x)/(1 - x*exp(t/(1 - x)))).
%F A211235 Read as a triangle: T(n,k) = Sum_{i = 1..k} binomial(n-i,k-i)*i^(n-k) for 1 <= k <= n.
%F A211235 n-th row polynomial of triangle: Sum_{i = 0..n-1} x^i*(x + i)^(n-i). (End)
%e A211235 Array begins:
%e A211235   1,       2,   3,   4,    5,     6, ... A000027
%e A211235   1,       4,  10,  20,   35,    56, ... A000292
%e A211235   1,       7,  27,  77,  182,   378, ... A005585
%e A211235   1,      12,  69, 272,  846,  2232, ... A101097
%e A211235   1,      21, 176, 936, 3750, 12342, ... A254681
%e A211235      A005126,
%e A211235   ...
%e A211235 Triangle begins:
%e A211235   1
%e A211235   1  2
%e A211235   1  4   3
%e A211235   1  7  10   4
%e A211235   1 12  27  20   5
%e A211235   1 21  69  77  35  6
%e A211235   1 38 176 272 182 56 7
%e A211235   ...
%p A211235 A211235 := (n, k) -> add(binomial(n-i, k-i)*i^(n-k), i = 1 .. k): for n from 1 to 10 do seq(A211235(n, k), k = 1 .. n) end do; # _Peter Bala_, Oct 27 2015
%t A211235 T[n_, k_] := Sum[Binomial[n-i, k-i] * i^(n-k), {i, 1, k}]; Table[T[n, k], {n,1,10}, {k,1,n}] //Flatten (* _Amiram Eldar_, Nov 30 2018 *)
%Y A211235 Cf. A008292, A211232-A211235.
%K A211235 nonn,tabl
%O A211235 1,3
%A A211235 _N. J. A. Sloane_, Apr 05 2012
%E A211235 Terms a(37)-a(55) added by _Peter Bala_, Oct 27 2015