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A246799
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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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1
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1, 7, 2, 34, 20, 3, 142, 128, 39, 4, 547, 668, 309, 64, 5, 2005, 3098, 1929, 604, 95, 6, 7108, 13304, 10434, 4384, 1040, 132, 7, 24604, 54128, 51258, 27064, 8600, 1644, 175, 8, 83653, 211592, 234966, 149536, 59630, 15252, 2443, 224, 9, 280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 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) = ((2*n+1)*3^(n+1) + 1)/4, for n >= 0.
T(n,n-1) = n*(3*n+4), for n >= 1.
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EXAMPLE
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Triangle starts:
1;
7, 2;
34, 20, 3;
142, 128, 39, 4;
547, 668, 309, 64, 5;
2005, 3098, 1929, 604, 95, 6;
7108, 13304, 10434, 4384, 1040, 132, 7;
24604, 54128, 51258, 27064, 8600, 1644, 175, 8;
83653, 211592, 234966, 149536, 59630, 15252, 2443, 224, 9;
280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 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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