A075497 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-2\,x + 16\,x^{2} - 20\,x^{3} + 8\,x^{4} - n + 8\,n x - 18\,n x^{2} + 28\,n x^{3} - 8\,n x^{4} - 4\,n^{2} x + 12\,n^{2} x^{2} - 12\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{3} - 8\,x^{4} + 8\,x^{5} + 2\,n x^{2} - 4\,n x^{3} + 12\,n x^{4} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 + 8\,x - 12\,x^{2} + 4\,x^{3} + 2\,n - 8\,n x + 10\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - 4\,x^{3} - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	7	14	4						25
3	13	68	52	8					141
4	21	220	460	184	16				901
5	31	550	2420	2704	656	32			6393
6	43	1162	8960	22064	14896	2400	64		49589
7	57	2184	26264	117264	181776	79680	9024	128	416377

All 15 rows

Technical details

Node id: n-c4d48ddc8d123a0e · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-820527166e270679

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:  1;
  1:  3,    2;
  2:  7,   14,      4;
  3: 13,   68,     52,       8;
  4: 21,  220,    460,     184,      16;
  5: 31,  550,   2420,    2704,     656,       32;
  6: 43, 1162,   8960,   22064,   14896,     2400,       64;
  7: 57, 2184,  26264,  117264,  181776,    79680,     9024,     128;
  8: 73, 3768,  65352,  466032, 1354416,  1410240,   422080,   34688,    256;
  9: 91, 6090, 144120, 1507968, 7169232, 14457120, 10542400, 2240768, 135424, 512;

%F Formula

The row polynomials satisfy (n-1+2*x)*P(n,x) = -(n+(4*n^2-8*n+2)*x+(4*n^3-12*n^2+18*n-16)*x^2+(12*n^2-28*n+20)*x^3+(8*n-8)*x^4)*P(n-2,x) + (2*n*x^2+(4*n^2-4*n+4)*x^3+(12*n-8)*x^4+8*x^5)*d/dx P(n-2,x) + (2*n-1+(4*n^2-8*n+8)*x+(10*n-12)*x^2+4*x^3)*P(n-1,x) - ((2*n-2)*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (4*n^2 - 8*n + 2)*x + (4*n^3 - 12*n^2 + 18*n - 16)*x^2 + (12*n^2 - 28*n + 20)*x^3 + (8*n - 8)*x^4)*P[n - 2, x] + (2*n*x^2 + (4*n^2 - 4*n + 4)*x^3 + (12*n - 8)*x^4 + 8*x^5)*D[P[n - 2, x], x] + (2*n - 1 + (4*n^2 - 8*n + 8)*x + (10*n - 12)*x^2 + 4*x^3)*P[n - 1, x] - ((2*n - 2)*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + (4*n^2 - 8*n + 2)*x + (4*n^3 - 12*n^2 + 18*n - 16)*x^2 + (12*n^2 - 28*n + 20)*x^3 + (8*n - 8)*x^4)*v[i - 2] + (2*n*x^2 + (4*n^2 - 4*n + 4)*x^3 + (12*n - 8)*x^4 + 8*x^5)*deriv(v[i - 2], x) + (2*n - 1 + (4*n^2 - 8*n + 8)*x + (10*n - 12)*x^2 + 4*x^3)*v[i - 1] - ((2*n - 2)*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*x)); 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 reversal of Stirling2 triangle with scaled diagonals (powers of 2).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Stirling2 triangle with scaled diagonals (powers of 2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Stirling2 triangle with scaled diagonals (powers of 2).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Stirling2 triangle with scaled diagonals (powers of 2).?
    
    ↑ Liu–Wang theorem (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.