A112007 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,n x^{2} - n^{2} + 4\,n^{2} x + 3\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(n x^{2} - n x^{3} - n^{2} x + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 2\,x + n - 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 0\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	8	2						10
3	58	44	3					105
4	444	656	156	4				1260
5	3708	8800	4356	456	5			17325
6	33984	116280	96360	22440	1200	6		270270
7	341136	1570608	1932060	783200	99750	2964	7	4729725

All 15 rows

Technical details

Node id: n-5d6fe78c6ecdb1ab · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3dcce632d1612cce

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        0;
  1:        1;
  2:        8,         2;
  3:       58,        44,         3;
  4:      444,       656,       156,         4;
  5:     3708,      8800,      4356,       456,         5;
  6:    33984,    116280,     96360,     22440,      1200,        6;
  7:   341136,   1570608,   1932060,    783200,     99750,     2964,       7;
  8:  3733920,  22052592,  37320192,  23062000,   5312500,   403560,    7028,     8;
  9: 44339040, 324373824, 716714712, 621429536, 223825000, 31956960, 1531936, 16208, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(n^3+n^2+(2*n^3-4*n^2)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (n^2*x-n*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2+n-2+(2*n^2-4*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (n^2*x - n*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + n - 2 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (n^2*x - n*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + n - 2 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A163936 → D

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.