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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.
3
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, 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, 45, 8, 462, 168, 30, 350, 195, 60, 6, 0, 1, 55, 9, 750, 252, 42, 1050, 560, 180, 24, 1, 0
OFFSET
0,8
LINKS
Eric Weisstein's World of Mathematics, Nørlund Polynomial.
FORMULA
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.
T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).
EXAMPLE
Array starts:
m\j| 0 1 2 3 4 5 6 7 8 9 10
---|-----------------------------------------------------------------------
m=0| 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
m=1| 1, 1, 1, 1, 3, 2, 1, 7, 12, 6, 1
m=2| 1, 3, 2, 7, 12, 6, 15, 50, 60, 24, 31
m=3| 1, 6, 3, 25, 30, 12, 90, 195, 180, 60, 301
m=4| 1, 10, 4, 65, 60, 20, 350, 560, 420, 120, 1701
m=5| 1, 15, 5, 140, 105, 30, 1050, 1330, 840, 210, 6951
m=6| 1, 21, 6, 266, 168, 42, 2646, 2772, 1512, 336, 22827
m=7| 1, 28, 7, 462, 252, 56, 5880, 5250, 2520, 504, 63987
m=8| 1, 36, 8, 750, 360, 72, 11880, 9240, 3960, 720, 159027
m=9| 1, 45, 9, 1155, 495, 90, 22275, 15345, 5940, 990, 359502
.
m\j| ... 11 12 13 14
---|-----------------------------------------
m=0| ..., 0, 0, 0, 0, ... [A000007]
m=1| ..., 15, 50, 60, 24, ... [A028246]
m=2| ..., 180, 390, 360, 120, ... [A053440]
m=3| ..., 1050, 1680, 1260, 360, ... [A294032]
m=4| ..., 4200, 5320, 3360, 840, ...
m=5| ..., 13230, 13860, 7560, 1680, ...
m=6| ..., 35280, 31500, 15120, 3024, ...
m=7| ..., 83160, 64680, 27720, 5040, ...
m=8| ..., 178200, 122760, 47520, 7920, ...
m=9| ..., 353925, 218790, 77220, 11880, ...
.
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.
.
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.
MAPLE
A293617 := proc(m, n, k) option remember:
if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293617(m, n-1, k) + k*A293617(m, n-1, k-1) + A293617(m-1, n, k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
# Sample uses:
A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);
MATHEMATICA
T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n, 0, 6}, {k, 0, n}]]]
A293617Row[m_, n_] := Table[T[m, n, k], {k, 0, n}];
(* Sample use: *)
A293926Row[n_] := A293617Row[n, n];
CROSSREFS
A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
Cf. A293616.
Sequence in context: A323222 A125104 A098157 * A165253 A336541 A317659
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Oct 20 2017
STATUS
approved