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%I A293616 #21 Feb 16 2025 08:33:51
%S A293616 1,1,0,1,1,0,1,3,0,0,1,6,0,1,0,1,10,0,7,1,0,1,15,0,25,4,0,0,1,21,0,65,
%T A293616 10,0,1,0,1,28,0,140,20,0,15,4,0,1,36,0,266,35,0,90,30,1,0,1,45,0,462,
%U A293616 56,0,350,120,5,0,0,1,55,0,750,84,0,1050,350,15,0,1,0
%N A293616 Array of generalized Eulerian number triangles read by ascending antidiagonals, with m >= 0, n >= 0 and 0 <= k <= n.
%H A293616 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html">Nielsen Generalized Polylogarithm</a>.
%F A293616 T(m, n, k) = (k + m)*T(m, n-1, k) + (n - k)*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k < 0 or k > n; and T(m, 0, k) = 0^k.
%F A293616 Let h(m, n) = x^(-m)*(1 - x)^(n + m)*PolyLog(-n - m, m, x) and p(m, n) the polynomial given by the expansion of h(m, n) after replacing log(1 - x) by 0. Then T(m, n, k) is the k-th coefficient of p(m, n) for 0 <= k < n.
%e A293616 Array starts:
%e A293616 m\j| 0 1 2 3 4 5 6 7 8 9 10 11 12
%e A293616 ---|----------------------------------------------------------------------------
%e A293616 m=0| 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A293616 m=1| 1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, ...
%e A293616 m=2| 1, 3, 0, 7, 4, 0, 15, 30, 5, 0, 31, 146, 91, ...
%e A293616 m=3| 1, 6, 0, 25, 10, 0, 90, 120, 15, 0, 301, 896, 406, ...
%e A293616 m=4| 1, 10, 0, 65, 20, 0, 350, 350, 35, 0, 1701, 3696, 1316, ...
%e A293616 m=5| 1, 15, 0, 140, 35, 0, 1050, 840, 70, 0, 6951, 11886, 3486, ...
%e A293616 m=6| 1, 21, 0, 266, 56, 0, 2646, 1764, 126, 0, 22827, 32172, 8022, ...
%e A293616 m=7| 1, 28, 0, 462, 84, 0, 5880, 3360, 210, 0, 63987, 76692, 16632, ...
%e A293616 m=8| 1, 36, 0, 750, 120, 0, 11880, 5940, 330, 0, 159027, 165792, 31812, ...
%e A293616 m=9| 1, 45, 0, 1155, 165, 0, 22275, 9900, 495, 0, 359502, 331617, 57057, ...
%e A293616 A000217, A001296,A000292,A001297,A027789,A000332,A001298,A293610,A293611, ...
%e A293616 .
%e A293616 m\j| ... 13 14 15 16 17 18 19 20
%e A293616 ---|----------------------------------------------------------------
%e A293616 m=0| ..., 0, 0, 0, 0, 0, 0, 0, 0, ... [A000007]
%e A293616 m=1| ..., 1, 0, 1, 26, 66, 26, 1, 0, ... [A173018]
%e A293616 m=2| ..., 6, 0, 63, 588, 868, 238, 7, 0, ... [A062253]
%e A293616 m=3| ..., 21, 0, 966, 5376, 5586, 1176, 28, 0, ... [A062254]
%e A293616 m=4| ..., 56, 0, 7770, 30660, 24570, 4200, 84, 0, ... [A062255]
%e A293616 m=5| ..., 126, 0, 42525, 129780, 84630, 12180, 210, 0, ...
%e A293616 m=6| ..., 252, 0, 179487, 446292, 245322, 30492, 462, 0, ...
%e A293616 m=7| ..., 462, 0, 627396, 1315776, 625086, 68376, 924, 0, ...
%e A293616 m=8| ..., 792, 0, 1899612, 3444012, 1440582, 140712, 1716, 0, ...
%e A293616 m=9| ..., 1287, 0, 5135130, 8198190, 3063060, 270270, 3003, 0, ...
%e A293616 A000389, A112494, A293612, A293613,A293614,A000579.
%e A293616 .
%e A293616 The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A062253, T(4, 2) is row 2 of A062255 (which is [65, 20, 0]) and T(4, 2, 1) = 20.
%p A293616 A293616 := proc(m, n, k) option remember:
%p A293616 if m = 0 then m^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
%p A293616 (k+m)*A293616(m,n-1,k) + (n-k)*A293616(m,n-1,k-1) + A293616(m-1,n,k) fi end:
%p A293616 for m in [$0..4] do for n in [$0..6] do print(seq(A293616(m, n, k), k=0..n)) od od;
%p A293616 # Sample uses:
%p A293616 A001298 := n -> A293616(n, 4, 0): A293614 := n -> A293616(n, 5, 3):
%p A293616 # Flatten:
%p A293616 a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
%p A293616 while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
%p A293616 seq(A293616(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);
%t A293616 GenEulerianRow[0, n_] := Table[If[n==0 && j==0,1,0], {j,0,n}];
%t A293616 GenEulerianRow[m_, n_] := If[n==0,{1},Join[CoefficientList[x^(-m) (1 - x)^(n+m)
%t A293616 PolyLog[-n-m, m, x] /. Log[1-x] -> 0, x], {0}]];
%t A293616 (* Sample use: *)
%t A293616 A173018Row[n_] := GenEulerianRow[1, n]; Table[A173018Row[n], {n, 0, 6}]
%Y A293616 A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A000292(n) = T(n, 2, 1),
%Y A293616 A001297(n) = T(n, 3, 0), A027789(n) = T(n, 3, 1), A000332(n) = T(n, 3, 2),
%Y A293616 A001298(n) = T(n, 4, 0), A293610(n) = T(n, 4, 1), A293611(n) = T(n, 4, 2),
%Y A293616 A000389(n) = T(n, 4, 3), A112494(n) = T(n, 5, 0), A293612(n) = T(n, 5, 1),
%Y A293616 A293613(n) = T(n, 5, 2), A293614(n) = T(n, 5, 3), A000579(n) = T(n, 5, 4),
%Y A293616 A144969(n) = T(n, 6, 0), A000580(n) = T(n, 6, 5), A000295(n) = T(1, n, 1),
%Y A293616 A000460(n) = T(1, n, 2), A000498(n) = T(1, n, 3), A000505(n) = T(1, n, 4),
%Y A293616 A000514(n) = T(1, n, 5), A001243(n) = T(1, n, 6), A001244(n) = T(1, n, 7),
%Y A293616 A126646(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
%Y A293616 Cf. A173018, A062253, A062254, A062255, A293609.
%K A293616 nonn,tabl
%O A293616 0,8
%A A293616 _Peter Luschny_, Oct 14 2017