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%I #16 Dec 18 2023 14:04:58
%S 1,1,0,1,1,0,1,3,1,0,1,6,2,1,0,1,10,3,7,3,0,1,15,4,25,12,2,0,1,21,5,
%T 65,30,6,1,0,1,28,6,140,60,12,15,7,0,1,36,7,266,105,20,90,50,12,0,1,
%U 45,8,462,168,30,350,195,60,6,0,1,55,9,750,252,42,1050,560,180,24,1,0
%N Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NorlundPolynomial.html">Nørlund Polynomial</a>.
%F T(m,n,k) = (k + m)*T(m, n-1, k) + 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 T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).
%e Array starts:
%e m\j| 0 1 2 3 4 5 6 7 8 9 10
%e ---|-----------------------------------------------------------------------
%e m=0| 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
%e m=1| 1, 1, 1, 1, 3, 2, 1, 7, 12, 6, 1
%e m=2| 1, 3, 2, 7, 12, 6, 15, 50, 60, 24, 31
%e m=3| 1, 6, 3, 25, 30, 12, 90, 195, 180, 60, 301
%e m=4| 1, 10, 4, 65, 60, 20, 350, 560, 420, 120, 1701
%e m=5| 1, 15, 5, 140, 105, 30, 1050, 1330, 840, 210, 6951
%e m=6| 1, 21, 6, 266, 168, 42, 2646, 2772, 1512, 336, 22827
%e m=7| 1, 28, 7, 462, 252, 56, 5880, 5250, 2520, 504, 63987
%e m=8| 1, 36, 8, 750, 360, 72, 11880, 9240, 3960, 720, 159027
%e m=9| 1, 45, 9, 1155, 495, 90, 22275, 15345, 5940, 990, 359502
%e A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
%e .
%e m\j| ... 11 12 13 14
%e ---|-----------------------------------------
%e m=0| ..., 0, 0, 0, 0, ... [A000007]
%e m=1| ..., 15, 50, 60, 24, ... [A028246]
%e m=2| ..., 180, 390, 360, 120, ... [A053440]
%e m=3| ..., 1050, 1680, 1260, 360, ... [A294032]
%e m=4| ..., 4200, 5320, 3360, 840, ...
%e m=5| ..., 13230, 13860, 7560, 1680, ...
%e m=6| ..., 35280, 31500, 15120, 3024, ...
%e m=7| ..., 83160, 64680, 27720, 5040, ...
%e m=8| ..., 178200, 122760, 47520, 7920, ...
%e m=9| ..., 353925, 218790, 77220, 11880, ...
%e A293476,A293608,A293615,A052762, ...
%e .
%e 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 A053440, T(3, 2) is row 2 of A294032 (which is [25, 30, 12]) and T(3, 2, 1) = 30.
%e .
%e Remark: To adapt the sequences A028246 and A053440 to our enumeration use the exponential generating functions exp(x)/(1 - y*(exp(x) - 1)) and exp(x)*(2*exp(x) - y*exp(2*x) + 2*y*exp(x) - 1 - y)/(1 - y*(exp(x) - 1))^2 instead of those indicated in their respective entries.
%p A293617 := proc(m, n, k) option remember:
%p if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
%p (k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
%p for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
%p # Sample uses:
%p A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
%p # Flatten:
%p a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
%p while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
%p seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);
%t T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
%t For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
%t A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
%t (* Sample use: *)
%t A293926Row[n_] := A293617Row[n, n];
%Y A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
%Y A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
%Y A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
%Y A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
%Y A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
%Y A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
%Y A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
%Y A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
%Y A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
%Y A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
%Y Cf. A293616.
%K nonn,tabl
%O 0,8
%A _Peter Luschny_, Oct 20 2017
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