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, 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

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 15 2008

Keywords

• sign
• tabl
• easy

Comments

This is essentially a signed version of A088996. - Peter Bala, Jan 09 2017

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A048994, A088996.

Programs

Magma

A142070:= func< n,k | (-1)^(n-k)*(&+[(-1)^j*Binomial(j,n-k)*StirlingFirst(n+1,n-j+1): j in [0..n]]) >;
[A142070(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 24 2022

Maple

A142070 := proc(n,k)
    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 *)

PARI

row(n) = Vecrev(prod(j=1, n, (1+j)*x - j)); \\ Michel Marcus, Feb 24 2022

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))
flatten([[A142070(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 24 2022

Formula

T(n, k) = [x^k]( Product_{j=1..n} ((1+j)*x - j) ).

Sum_{k=0..n} T(n, k) = 1.

From G. C. Greubel, Feb 24 2022: (Start)

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)!. (End)