OFFSET
0,4
FORMULA
Let P_{n}(x) = Sum_{k=0..n} p_{n,k}(x) then
2^n*P_{n}(1/2) = A299502(n).
The general case: for fixed k the sequence P_{n}(k) with n >= 0 has the generating function ogf(k, x) = (1-2*(k+1)*x + (k^2+2*k-1)*x^2 - 2*(k-1)*x^3 + x^4)^(-1/2). The example section shows the start of this square array of sequences.
These sequences can be computed by the recurrence P(n,k) = ((2-n)*P(n-4,k)-(3-2*n)*(k-1)*P(n-3,k)+(k^2+2*k-1)*(1-n)*P(n-2,k)+(2*n-1)*(k+1)*P(n-1,k))/n with initial values 1, k+1, (k+1)^2+1 and (k+1)^3+4*k+2.
The partial polynomials p_{n,k}(x) reduce for x = 1 to A108625 (seen as a triangle).
EXAMPLE
The partial polynomials p_{n,k}(x) start:
[0] 1
[1] x, 1
[2] x^2, 2*x+1, 1
[3] x^3, 3*x^2+4*x, 3*x+2, 1
[4] x^4, 4*x^3+9*x^2, 6*x^2+12*x+1, 4*x+3, 1
[5] x^5, 5*x^4+16*x^3, 10*x^3+36*x^2+9*x, 10*x^2+24*x+3, 5*x+4, 1
.
The polynomials P_{n}(x) start:
[0] 1
[1] 1 + x
[2] 2 + 2*x + x^2
[3] 3 + 7*x + 3*x^2 + x^3
[4] 5 + 16*x + 15*x^2 + 4*x^3 + x^4
[5] 8 + 38*x + 46*x^2 + 26*x^3 + 5*x^4 + x^5
.
The triangle starts:
[0] 1
[1] 1, 1
[2] 2, 2, 1
[3] 3, 7, 3, 1
[4] 5, 16, 15, 4, 1
[5] 8, 38, 46, 26, 5, 1
[6] 13, 82, 141, 100, 40, 6, 1
[7] 21, 173, 381, 375, 185, 57, 7, 1
[8] 34, 352, 983, 1216, 820, 308, 77, 8, 1
[9] 55, 701, 2400, 3704, 3101, 1575, 476, 100, 9, 1'
.
The square array P_{n}(k) near k=0:
...... [k=-2] 1, -1, 2, -7, 17, -44, 125, -345, 958, -2707, ...
A182883 [k=-1] 1, 0, 1, -2, 1, -6, 7, -12, 31, -40, ...
A000045 [k=0] 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
A108626 [k=1] 1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, ...
A299501 [k=2] 1, 3, 10, 37, 145, 588, 2437, 10251, 43582, 186785, ...
...... [k=3] 1, 4, 17, 78, 377, 1886, 9655, 50220, 264223, 1402108, ...
MAPLE
CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
PrintPoly := p -> print(sort(expand(p), x, ascending)):
T := (n, k) -> x^(n-k)*binomial(n, k)*hypergeom([-k, k-n, k-n], [1, -n], 1/x):
P := [seq(add(simplify(T(n, k)), k=0..n), n=0..10)]:
seq(CoeffList(p), p in P); # seq(PrintPoly(p), p in P);
R := proc(n, k) option remember; # Recurrence
if n < 4 then return [1, k+1, (k+1)^2+1, (k+1)^3+4*k+2][n+1] fi; ((2-n)*R(n-4, k)-
(3-2*n)*(k-1)*R(n-3, k)+(k^2+2*k-1)*(1-n)*R(n-2, k)+(2*n-1)*(k+1)*R(n-1, k))/n end:
for k from -2 to 3 do lprint(seq(R(n, k), n=0..9)) od;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Feb 11 2018
STATUS
approved