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A242598 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. 2

%I #42 Oct 23 2014 20:48:47

%S 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,

%T 386,37,1,2,140,3330,7862,4306,772,50,1,2,204,10498,44150,40146,11972,

%U 1394,65,1,2,285,31234,227858,330450,153722,28610,2333,82,1,2,385,88834,1102658,2480850,1728722,482210,61133,3682,101,1

%N 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.

%C 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.

%F T(n,1) = n*(2*n+1)*(n+1)/6 for n > 0.

%F T(n,n-1) = n^2 + 1 for n > 0.

%F Rows sum to SUM{k=0..n} A138911(k).

%e 1;

%e 2, 1;

%e 2, 5, 1;

%e 2, 14, 10, 1;

%e 2, 30, 58, 17, 1;

%e 2, 55, 258, 167, 26, 1;

%e 2, 91, 978, 1247, 386, 37, 1;

%e 2, 140, 3330, 7862, 4306, 772, 50, 1;

%e 2, 204, 10498, 44150, 40146, 11972, 1394, 65, 1;

%e 2, 285, 31234, 227858, 330450, 153722, 28610, 2333, 82, 1;

%e 2, 385, 88834, 1102658, 2480850, 1728722, 482210, 61133, 3682, 101, 1

%o (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,", "))))

%Y Cf. A000330, A002522, A138911, A248826.

%K nonn,tabl

%O 0,2

%A _Derek Orr_, Oct 15 2014

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)