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A242598
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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.
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2
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1, 2, 1, 2, 5, 1, 2, 14, 10, 1, 2, 30, 58, 17, 1, 2, 55, 258, 167, 26, 1, 2, 91, 978, 1247, 386, 37, 1, 2, 140, 3330, 7862, 4306, 772, 50, 1, 2, 204, 10498, 44150, 40146, 11972, 1394, 65, 1, 2, 285, 31234, 227858, 330450, 153722, 28610, 2333, 82, 1, 2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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T(n,1) = n*(2*n+1)*(n+1)/6 for n > 0.
T(n,n-1) = n^2 + 1 for n > 0.
Rows sum to SUM{k=0..n} A138911(k).
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EXAMPLE
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1;
2, 1;
2, 5, 1;
2, 14, 10, 1;
2, 30, 58, 17, 1;
2, 55, 258, 167, 26, 1;
2, 91, 978, 1247, 386, 37, 1;
2, 140, 3330, 7862, 4306, 772, 50, 1;
2, 204, 10498, 44150, 40146, 11972, 1394, 65, 1;
2, 285, 31234, 227858, 330450, 153722, 28610, 2333, 82, 1;
2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1
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PROG
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(PARI) for(n=0, 20, for(k=0, n, if(!k, if(n, print1(2, ", ")); if(!n, print1(1, ", "))); if(k, print1(sum(i=1, n, (k^(i-k)*i*binomial(i, k)))/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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