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A371898
Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.
3
1, 1, 1, 1, 4, 4, 1, 15, 48, 36, 1, 64, 504, 1008, 576, 1, 325, 5680, 22680, 31680, 14400, 1, 1956, 72060, 510480, 1304640, 1382400, 518400, 1, 13699, 1036224, 12233340, 50823360, 94046400, 79833600, 25401600, 1, 109600, 16798768, 318469536, 2017814400, 5794790400, 8346240000, 5893171200, 1625702400
OFFSET
0,5
FORMULA
T(n, k) = Sum_{i=k..n} A131689(i, k) * n! / (n-i)!.
T(n, k) = n! * k! * (Sum_{i=0..n-k} A048993(n-i, k) / i!).
T(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A320031(n, i).
Conjecture: E.g.f. of column k is exp(t) * t^k * k! / (Prod_{i=0..k} (1 - i*t)).
Conjecture: Sum_{k=0..n} (-1)^(n-k) * T(n, k) = A000166(n).
T(n, k) = A371766(n, k) * A371767(n, k). - Peter Luschny, Apr 14 2024
EXAMPLE
Lower triangular array starts:
n\k : 0 1 2 3 4 5 6 7
==========================================================================
0 : 1
1 : 1 1
2 : 1 4 4
3 : 1 15 48 36
4 : 1 64 504 1008 576
5 : 1 325 5680 22680 31680 14400
6 : 1 1956 72060 510480 1304640 1382400 518400
7 : 1 13699 1036224 12233340 50823360 94046400 79833600 25401600
etc.
MATHEMATICA
T[n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*HypergeometricPFQ[{1, -n}, {}, -j], {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Apr 12 2024 *)
PROG
(PARI) T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))
CROSSREFS
Cf. A000012 (column 0), A007526 (column 1), A001044 (main diagonal).
Sequence in context: A145902 A124028 A123966 * A079507 A098364 A116866
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Apr 11 2024
STATUS
approved