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A246798
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Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x+3)^k.
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0
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1, -5, 2, 22, -16, 3, -86, 92, -33, 4, 319, -448, 237, -56, 5, -1139, 1982, -1383, 484, -85, 6, 3964, -8224, 7122, -3296, 860, -120, 7, -13532, 32600, -33702, 19384, -6700, 1392, -161, 8, 45517, -124864, 150006, -103088, 44330, -12216, 2107, -208, 9, -151313, 465626, -637314, 509272, -261850, 89844, -20573, 3032, -261, 10
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OFFSET
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0,2
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COMMENTS
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Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*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.
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LINKS
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FORMULA
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T(n,0) = (1-(4*n+5)*(-3)^(n+1))/16, for n >= 0.
T(n,n-1) = -n*(3*n+2), for n >= 1.
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EXAMPLE
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Triangle starts:
1;
-5, 2;
22, -16, 3;
-86, 92, -33, 4;
319, -448, 237, -56, 5;
-1139, 1982, -1383, 484, -85, 6;
3964, -8224, 7122, -3296, 860, -120, 7;
-13532, 32600, -33702, 19384, -6700, 1392, -161, 8;
45517, -124864, 150006, -103088, 44330, -12216, 2107, -208, 9;
-151313, 465626, -637314, 509272, -261850, 89844, -20573, 3032, -261, 10;
...
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PROG
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(PARI) T(n, k) = (k+1)*sum(i=0, n-k, (-3)^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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