A249267 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-2*(-1)^k)^k.

1, -1, 1, -13, 5, 1, 75, -31, -5, 1, 987, -383, -77, 9, 1, -10565, 4177, 803, -111, -9, 1, -187397, 73489, 14483, -1871, -189, 13, 1, 2962811, -1164335, -228109, 30049, 2891, -239, -13, 1, 67151483, -26365999, -5179405, 676961, 66731, -5167, -349, 17, 1

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

1;
-1,               1;
-13,              5,        1;
75,             -31,       -5,      1;
987,           -383,      -77,      9,     1;
-10565,        4177,      803,   -111,    -9,     1;
-187397,      73489,    14483,  -1871,  -189,    13,    1;
2962811,   -1164335,  -228109,  30049,  2891,  -239,  -13,  1;
67151483, -26365999, -5179405, 676961, 66731, -5167, -349, 17, 1;

Prog

(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, -2), ", ")))

Crossrefs

Cf. A248976.

Sequence in context: A226376 A222165 A277125 * A123172 A010218 A107833

Adjacent sequences: A249264 A249265 A249266 * A249268 A249269 A249270

Keyword

sign,tabl

Author

Derek Orr, Oct 23 2014

Status

approved