A248830 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-2k)^k for 0 <= k <= n.

1, 3, 1, 3, 9, 1, 3, 45, 19, 1, 3, 173, 211, 33, 1, 3, 573, 1811, 633, 51, 1, 3, 1725, 13331, 9273, 1491, 73, 1, 3, 4861, 88595, 115113, 32851, 3013, 99, 1, 3, 13053, 547347, 1276329, 606291, 92613, 5475, 129, 1, 3, 33789, 3201555, 13033641, 9896019, 2360613, 223203, 9201, 163, 1, 3, 84989, 17947155, 125008041, 147521619, 52760613, 7480803, 479601, 14563, 201, 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-2)^1 + A_2*(x-4)^2 + ... + A_n*(x-2n)^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;
3,     1;
3,     9,       1;
3,    45,      19,        1;
3,   173,     211,       33,       1;
3,   573,    1811,      633,      51,       1;
3,  1725,   13331,     9273,    1491,      73,      1;
3,  4861,   88595,   115113,   32851,    3013,     99,    1;
3, 13053,  547347,  1276329,  606291,   92613,   5475,  129,   1;
3, 33789, 3201555, 13033641, 9896019, 2360613, 223203, 9201, 163, 1;
		

Crossrefs

Cf. A036826, A058331, A248829, A242598.

Programs

PARI

for(n=0,10,for(k=0,n,if(!k,if(n,print1(3,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,((2*k)^(i-k)*i*binomial(i,k)))/k,", "))))

Formula

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

T(n,1) = 2^n*(n^2-2*n+3)-3 for n > 0.