|
|
A142070
|
|
Triangle T(n,k) read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (i+1)*x-i in row n>=0 and column 0<=k<=n.
|
|
1
|
|
|
1, -1, 2, 2, -7, 6, -6, 29, -46, 24, 24, -146, 329, -326, 120, -120, 874, -2521, 3604, -2556, 720, 720, -6084, 21244, -39271, 40564, -22212, 5040, -5040, 48348, -197380, 444849, -598116, 479996, -212976, 40320, 40320, -432144, 2014172, -5335212, 8788569, -9223012, 6023772, -2239344, 362880
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = [x^k]( Product_{j=1..n} ((1+j)*x - j) ).
Sum_{k=0..n} T(n, k) = 1.
T(n, k) = (-1)^(n-k) * Sum_{j=0..n} (-1)^j*binomial(j,n-k)*Stirling1(n+1, n-j+1).
T(n, k) = Sum_{j=0..k} (-1)^j*binomial(j+n-k,n-k)*Stirling1(n+1, k-j+1).
T(n, 0) = (-1)^n * n!.
T(n, n) = (n+1)!.
|
|
EXAMPLE
|
Triangle begins as:
1;
-1, 2;
2, -7, 6;
-6, 29, -46, 24;
24, -146, 329, -326, 120;
-120, 874, -2521, 3604, -2556, 720;
720, -6084, 21244, -39271, 40564, -22212, 5040;
-5040, 48348, -197380, 444849, -598116, 479996, -212976, 40320;
40320, -432144, 2014172, -5335212, 8788569, -9223012, 6023772, -2239344, 362880;
|
|
MAPLE
|
local x, i ;
mul( (i+1)*x-i, i=1..n) ;
expand(%) ;
coeff(%, x, k) ;
end proc:
|
|
MATHEMATICA
|
(* First program *)
p[x_, n_]:= Product[(i+1)*x - i, {i, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= Sum[(-1)^j*Binomial[j+n-k, n-k]*StirlingS1[n+1, k-j+1], {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 24 2022 *)
|
|
PROG
|
(Magma)
A142070:= func< n, k | (-1)^(n-k)*(&+[(-1)^j*Binomial(j, n-k)*StirlingFirst(n+1, n-j+1): j in [0..n]]) >;
(Sage)
def A142070(n, k): return (-1)^(n-k)*sum(binomial(j+n-k, n-k)*stirling_number1(n+1, k-j+1) for j in (0..k))
(PARI) row(n) = Vecrev(prod(j=1, n, (1+j)*x - j)); \\ Michel Marcus, Feb 24 2022
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|