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.

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

Offset: 0

Author

Derek Orr, Oct 15 2014

Keywords

• nonn
• tabl

Comments

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.

Examples

			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
		

Crossrefs

Cf. A000330, A002522, A138911, A248826.

Programs

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,", "))))

Formula

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).