A249268 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} x^k = Sum_{k=0..n} A_k*(x+3*(-1)^k)^k.

1, 4, 1, -23, -5, 1, -320, -86, 10, 1, 4297, 1102, -152, -11, 1, 92020, 24187, -3122, -281, 16, 1, -1922207, -502151, 66133, 5659, -389, -17, 1, -55746464, -14601740, 1908316, 167254, -10784, -584, 22, 1, 1589338993, 415992316, -54490040, -4745234, 312406, 16048, -734, -23, 1

Offset: 0

Author

Derek Orr, Oct 23 2014

Keywords

• sign
• tabl

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+3)^0 + A_1*(x-3)^1 + A_2*(x+3)^2 + A_3*(x-3)^3 + ... + A_n*(x+3*(-1)^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

			Triangle starts:
1;
4,                  1;
-23,               -5,         1;
-320,             -86,        10,        1;
4297,            1102,      -152,      -11,      1;
92020,          24187,     -3122,     -281,     16,     1;
-1922207,     -502151,     66133,     5659,   -389,   -17,    1;
-55746464,  -14601740,   1908316,   167254, -10784,  -584,   22,   1;
1589338993, 415992316, -54490040, -4745234, 312406, 16048, -734, -23, 1;
...
		

Crossrefs

Cf. A248975, A248977, A248811.

Programs

PARI

a(n, j, L)=if(j==n, return(1)); if(j!=n, return(1-sum(i=1, n-j, (-L)^i*(-1)^(i*j)*binomial(i+j, i)*a(n, i+j, L))))
for(n=0, 10, for(j=0, n, print1(a(n, j, 3), ", ")))