A074248 Triangle of coefficients, read by rows of (2n+1) terms, where the n-th row forms a polynomial in x, P(n,x), of degree 2n and satisfies: P(n,x) = [Sum_{k=1..n} 1/(k + x + x^2)]*[Product_{k=1..n} (k + x + x^2)].

1, 3, 2, 2, 11, 12, 15, 6, 3, 50, 70, 100, 64, 42, 12, 4, 274, 450, 705, 570, 440, 200, 90, 20, 5, 1764, 3248, 5453, 5110, 4410, 2526, 1360, 480, 165, 30, 6, 13068, 26264, 46571, 48454, 45437, 30128, 18347, 8162, 3395, 980, 273, 42, 7

Offset

1,2

Comments

These polynomials have zeros at complex z_k such that real(z_k) = -1/2 for all 0 < k < (2n-1), n > 1. A pair of zeros that are complex rationals occur at n = 2k(k+1) and have the values z = -1/2 +- (2k+1)/2*i for k > 0. P(n,0) = Stirling numbers of the first kind and the product of all the zeros of P(n,x) equals P(n,0)/n!.

Links

Vincenzo Librandi, Rows n = 1..32, flattened

Example

P(1,x) = 1,
P(2,x) = 3 + 2x + 2x^2,
P(3,x) = 11 + 12x + 15x^2 + 6x^3 + 3x^4,
P(4,x) = 50 + 70x + 100x^2 + 64x^3 + 42x^4 + 12x^5 + 4x^6,
P(5,x) = 274 + 450x + 705x^2 + 570x^3 + 440x^4 + 200x^5 + 90x^6 + 20x^7 + 5x^8,
P(6,x) = 1764 + 3248x + 5453x^2 + 5110x^3 + 4410x^4 + 2526x^5 + 1360x^6 + 480x^7 + 165x^8 + 30x^9 + 6x^10,
P(7,x) = 13068 + 26264x + 46571x^2 + 48454x^3 + 45437x^4 + 30128x^5 + 18347x^6 + 8162x^7 + 3395x^8 + 980x^9 + 273x^10 + 42x^11 + 7x^12.

Mathematica

p[n_, x_] := Sum[1/(k + x + x^2), {k, 1, n}]*Product[k + x + x^2, {k, 1, n}]; row[n_] := CoefficientList[ Series[p[n, x], {x, 0, 2*n-2}], x]; Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Aug 16 2013 *)

Crossrefs

Sequence in context: A100804 A143175 A368557 * A266004 A206703 A122101

Adjacent sequences: A074245 A074246 A074247 * A074249 A074250 A074251

Keyword

easy,nice,nonn,tabf

Author

Paul D. Hanna, Sep 20 2002

Extensions

Keyword tabf by Michel Marcus, Aug 06 2017

Status

approved