A248826 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+k)^k for 0 <= k <= n.

1, 0, 1, 0, -3, 1, 0, 6, -8, 1, 0, -10, 40, -15, 1, 0, 15, -160, 135, -24, 1, 0, -21, 560, -945, 336, -35, 1, 0, 28, -1792, 5670, -3584, 700, -48, 1, 0, -36, 5376, -30618, 32256, -10500, 1296, -63, 1, 0, 45, -15360, 153090, -258048, 131250, -25920, 2205, -80, 1, 0, -55, 42240, -721710, 1892352, -1443750, 427680, -56595, 3520, -99, 1

Offset: 0

Author

Derek Orr, Oct 15 2014

Keywords

• sign
• tabl

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+0)^0 + A_1*(x+1)^1 + A_2*(x+2)^2 + ... + A_n*(x+n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Examples

			1;
0,   1;
0,  -3,      1;
0,   6,     -8,       1;
0, -10,     40,     -15,       1;
0,  15,   -160,     135,     -24,        1;
0, -21,    560,    -945,     336,      -35,      1;
0,  28,  -1792,    5670,   -3584,      700,    -48,      1;
0, -36,   5376,  -30618,   32256,   -10500,   1296,    -63,    1;
0,  45, -15360,  153090, -258048,   131250, -25920,   2205,  -80,   1;
0, -55,  42240, -721710, 1892352, -1443750, 427680, -56595, 3520, -99, 1;
		

Crossrefs

Cf. A001789, A000217.

Programs

PARI

for(n=0,20,for(k=0,n,if(!k,if(n,print1(0,", "));if(!n,print1(1,", ")));if(k,print1(-sum(i=1,n,((-k)^(i-k-1)*i*binomial(i,k))),", "))))

Formula

T(n,1) = n*(n+1)*(-1)^(n+1)/2 for n > 0.

T(n,2) = Binomial(n+1,3)*2^(n-2)*(-1)^n for n > 1.

T(n,n-1) = 1 - n^2 for n > 0.

T(n,n-2) = (1/2)*n*(n-2)^2*(n+1) for n > 1.