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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.
3
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
0,2
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.
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).
EXAMPLE
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;
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 *)
PROG
(PARI) for(n=0, 20, for(k=0, n, print1(sum(i=0, n, ((-3)^(i-k)* binomial(i, k)) ), ", ")))
(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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Derek Orr, Oct 14 2014
STATUS
approved