A248811 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+3)^k for 0 <= k <= n.

1, -2, 1, 7, -5, 1, -20, 22, -8, 1, 61, -86, 46, -11, 1, -182, 319, -224, 79, -14, 1, 547, -1139, 991, -461, 121, -17, 1, -1640, 3964, -4112, 2374, -824, 172, -20, 1, 4921, -13532, 16300, -11234, 4846, -1340, 232, -23, 1, -14762, 45517, -62432, 50002, -25772, 8866, -2036, 301, -26, 1, 44287, -151313, 232813, -212438, 127318, -52370, 14974, -2939, 379, -29, 1

Offset: 0

Author

Derek Orr, Oct 14 2014

Keywords

• sign
• tabl

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+3)^0 + A_1*(x+3)^1 + A_2*(x+3)^2 + ... + A_n*(x+3)^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;
      -2,       1;
       7,      -5,      1;
     -20,      22,     -8,       1;
      61,     -86,     46,     -11,      1;
    -182,     319,   -224,      79,    -14,      1;
     547,   -1139,    991,    -461,    121,    -17,     1;
   -1640,    3964,  -4112,    2374,   -824,    172,   -20,     1;
    4921,  -13532,  16300,  -11234,   4846,  -1340,   232,   -23,   1;
  -14762,   45517, -62432,   50002, -25772,   8866, -2036,   301, -26,   1;
   44287, -151313, 232813, -212438, 127318, -52370, 14974, -2939, 379, -29, 1;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Crossrefs

Cf. A193843, A077925, A191008, A014983.

Programs

Magma

[[(&+[(-3)^(j-k)*Binomial(j,k): j in [0..n]]): k in [0..n]]: n in [0..20]]; // G. C. Greubel, May 27 2018

Mathematica

T[n_, k_]:= Sum[(-3)^(j-k)*Binomial[j,k], {j,0,n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 27 2018 *)

PARI

for(n=0,20,for(k=0,n,print1(sum(i=0,n,((-3)^(i-k)* binomial(i, k)) ),", ")))

Formula

T(n,n-1) = -3*n + 1 for n > 0.

T(n,0) = A014983(n+1).

T(n,1) = (-1)^(n+1)*A191008(n-1).

Row n sums to A077925(n).