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A293617 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.

%I A293617 #17 Feb 16 2025 08:33:51
%S A293617 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 A293617 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 A293617 45,8,462,168,30,350,195,60,6,0,1,55,9,750,252,42,1050,560,180,24,1,0
%N A293617 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 A293617 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NorlundPolynomial.html">Nørlund Polynomial</a>.
%F A293617 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 A293617 T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).
%e A293617 Array starts:
%e A293617 m\j| 0   1  2     3       4       5       6       7       8       9      10
%e A293617 ---|-----------------------------------------------------------------------
%e A293617 m=0| 1,  0, 0,    0,      0,      0,      0,      0,      0,      0,      0
%e A293617 m=1| 1,  1, 1,    1,      3,      2,      1,      7,     12,      6,      1
%e A293617 m=2| 1,  3, 2,    7,     12,      6,     15,     50,     60,     24,     31
%e A293617 m=3| 1,  6, 3,   25,     30,     12,     90,    195,    180,     60,    301
%e A293617 m=4| 1, 10, 4,   65,     60,     20,    350,    560,    420,    120,   1701
%e A293617 m=5| 1, 15, 5,  140,    105,     30,   1050,   1330,    840,    210,   6951
%e A293617 m=6| 1, 21, 6,  266,    168,     42,   2646,   2772,   1512,    336,  22827
%e A293617 m=7| 1, 28, 7,  462,    252,     56,   5880,   5250,   2520,    504,  63987
%e A293617 m=8| 1, 36, 8,  750,    360,     72,  11880,   9240,   3960,    720, 159027
%e A293617 m=9| 1, 45, 9, 1155,    495,     90,  22275,  15345,   5940,    990, 359502
%e A293617    A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
%e A293617 .
%e A293617 m\j| ...      11      12      13      14
%e A293617 ---|-----------------------------------------
%e A293617 m=0| ...,      0,      0,      0,      0, ... [A000007]
%e A293617 m=1| ...,     15,     50,     60,     24, ... [A028246]
%e A293617 m=2| ...,    180,    390,    360,    120, ... [A053440]
%e A293617 m=3| ...,   1050,   1680,   1260,    360, ... [A294032]
%e A293617 m=4| ...,   4200,   5320,   3360,    840, ...
%e A293617 m=5| ...,  13230,  13860,   7560,   1680, ...
%e A293617 m=6| ...,  35280,  31500,  15120,   3024, ...
%e A293617 m=7| ...,  83160,  64680,  27720,   5040, ...
%e A293617 m=8| ..., 178200, 122760,  47520,   7920, ...
%e A293617 m=9| ..., 353925, 218790,  77220,  11880, ...
%e A293617          A293476,A293608,A293615,A052762, ...
%e A293617 .
%e A293617 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 A293617 .
%e A293617 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 A293617 := proc(m, n, k) option remember:
%p A293617 if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
%p A293617 (k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
%p A293617 for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
%p A293617 # Sample uses:
%p A293617 A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
%p A293617 # Flatten:
%p A293617 a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
%p A293617 while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
%p A293617 seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);
%t A293617 T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
%t A293617 For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
%t A293617 A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
%t A293617 (* Sample use: *)
%t A293617 A293926Row[n_] := A293617Row[n, n];
%Y A293617 A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
%Y A293617 A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
%Y A293617 A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
%Y A293617 A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
%Y A293617 A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
%Y A293617 A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
%Y A293617 A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
%Y A293617 A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
%Y A293617 A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
%Y A293617 A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
%Y A293617 Cf. A293616.
%K A293617 nonn,tabl
%O A293617 0,8
%A A293617 _Peter Luschny_, Oct 20 2017