A299499 Triangle read by rows, T(n,k) = [x^k] Sum_{k=0..n} p_{n,k}(x) where p_{n,k}(x) = x^k*binomial(n, k)*hypergeom([-k, k-n, k-n], [1, -n], 1/x), for 0 <= k <= n.

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 11, 16, 9, 4, 1, 26, 44, 34, 14, 5, 1, 63, 122, 111, 60, 20, 6, 1, 153, 341, 351, 225, 95, 27, 7, 1, 376, 940, 1103, 796, 400, 140, 35, 8, 1, 931, 2581, 3384, 2764, 1561, 651, 196, 44, 9, 1, 2317, 7064, 10224, 9304, 5915, 2772, 994, 264, 54, 10, 1

Offset

0,4

Links

Table of n, a(n) for n=0..65.

Formula

Let P_{n}(x) = Sum_{k=0..n} p_{n,k}(x) then

2^n*P_{n}(1/2) = A298611(n).

P_{n}(-1) = A182883(n), P_{n}(0) = A051286(n).

P_{n}( 1) = A108626(n), P_{n}(2) = A299443(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+2*k+4.

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] 1, x
[2] 1, 2*x+ 1,    x^2
[3] 1, 3*x+ 4,  3*x^2+ 2*x,      x^3
[4] 1, 4*x+ 9,  6*x^2+12*x+1,  4*x^3+ 3*x^2,       x^4
[5] 1, 5*x+16, 10*x^2+36*x+9, 10*x^3+24*x^2+3*x, 5*x^4+4*x^3, x^5
.
The polynomials P_{n}(x) start:
[0]   1
[1]   1 +    x
[2]   2 +  2*x +    x^2
[3]   5 +  5*x +  3*x^2 +    x^3
[4]  11 + 16*x +  9*x^2 +  4*x^3 +   x^4
[5]  26 + 44*x + 34*x^2 + 14*x^3 + 5*x^4 + x^5
.
The triangle starts:
[0]   1
[1]   1,    1
[2]   2,    2,    1
[3]   5,    5,    3,    1
[4]  11,   16,    9,    4,    1
[5]  26,   44,   34,   14,    5,   1
[6]  63,  122,  111,   60,   20,   6,   1
[7] 153,  341,  351,  225,   95,  27,   7,  1
[8] 376,  940, 1103,  796,  400, 140,  35,  8, 1
[9] 931, 2581, 3384, 2764, 1561, 651, 196, 44, 9, 1
.
The square array P_{n}(k) near k=0:
......  [k=-2] 1, -1,  2, -1,  -1,   10,  -25,    51,    -68,     41, ...
A182883 [k=-1] 1,  0,  1,  2,   1,    6,    7,    12,     31,     40, ...
A051286 [k=0]  1,  1,  2,  5,  11,   26,   63,   153,    376,    931, ...
A108626 [k=1]  1,  2,  5, 14,  41,  124,  383,  1200,   3799,  12122, ...
A299443 [k=2]  1,  3, 10, 35, 127,  474, 1807,  6999,  27436, 108541, ...
......  [k=3]  1,  4, 17, 74, 329, 1490, 6855, 31956, 150607, 716236, ...

Maple

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
PrintPoly := p -> print(sort(expand(p), x, ascending)):
T := (n, k) -> x^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+2*k+4][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;

Mathematica

nmax = 10;
p[n_, k_, x_] := x^k*Binomial[n, k]*HypergeometricPFQ[{-k, k-n, k-n}, {1, -n}, 1/x];
p[n_, x_] := Sum[p[n, k, x], {k, 0, n}];
Table[CoefficientList[p[n, x], x], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Feb 26 2018 *)

Crossrefs

Cf. A051286, A108625, A108626, A182883, A298611, A299443, A299500.

Sequence in context: A114292 A333724 A178518 * A190215 A190252 A141751

Adjacent sequences: A299496 A299497 A299498 * A299500 A299501 A299502

Keyword

nonn,tabl

Author

Peter Luschny, Feb 11 2018

Status

approved