A106800 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-x - 2\,x^{3} + n x^{3}\right)\,P_{n}(x) = \left(-4 - 9\,x - 4\,x^{2} - 3\,x^{3} + 6\,x^{5} + 2\,n + n x + 5\,n x^{3} - 6\,n x^{4} - 11\,n x^{5} + n^{2} x + n^{2} x^{3} + 8\,n^{2} x^{4} + 6\,n^{2} x^{5} - n^{3} x^{3} - 2\,n^{3} x^{4} - n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-1 - 2\,x + 2\,x^{3} + 3\,x^{4} + 4\,x^{5} + 2\,x^{6} - n x - 2\,n x^{2} - 2\,n x^{3} - 5\,n x^{4} - 7\,n x^{5} - 3\,n x^{6} + n^{2} x^{3} + 3\,n^{2} x^{4} + 3\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-2 + x + 2\,x^{2} + 2\,x^{3} + 8\,x^{4} - 2\,n x - 4\,n x^{3} - 8\,n x^{4} + 2\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + 3\,x^{3} + 4\,x^{4} + 2\,x^{5} - n x^{3} - 2\,n x^{4} - n x^{5}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1	1							1
2	1							1
3	2	2						4
4	5	10	3					18
5	15	46	30	4				95
6	52	218	228	76	5			579
7	203	1088	1593	896	180	6		3966
8	877	5752	10923	8816	3155	414	7	29944

All 15 rows

Technical details

Node id: n-1e338c79a9d4cfe9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-5f72f3d1614d5d16

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     1;
   3:     2,      2;
   4:     5,     10,      3;
   5:    15,     46,     30,      4;
   6:    52,    218,    228,     76,      5;
   7:   203,   1088,   1593,    896,    180,      6;
   8:   877,   5752,  10923,   8816,   3155,    414,     7;
   9:  4140,  32226,  75624,  80356,  42355,  10494,   938,    8;
  10: 21147, 190990, 535692, 710524, 500855, 187986, 33866, 2104, 9;

%F Formula

The row polynomials satisfy -(x-(n-2)*x^3)*P(n,x) = (2*n-4+(n^2+n-9)*x-4*x^2-(n^3-n^2-5*n+3)*x^3-(2*n^3-8*n^2+6*n)*x^4-(n^3-6*n^2+11*n-6)*x^5)*P(n-2,x) - (1+(n+2)*x+2*n*x^2-(n^2-2*n+2)*x^3-(3*n^2-5*n+3)*x^4-(3*n^2-7*n+4)*x^5-(n^2-3*n+2)*x^6)*d/dx P(n-2,x) - (2+(2*n-1)*x-2*x^2-(2*n^2-4*n+2)*x^3-(2*n^2-8*n+8)*x^4)*P(n-1,x) + (x+2*x^2-(n-3)*x^3-(2*n-4)*x^4-(n-2)*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n^2 + n - 9)*x - 4*x^2 - (n^3 - n^2 - 5*n + 3)*x^3 - (2*n^3 - 8*n^2 + 6*n)*x^4 - (n^3 - 6*n^2 + 11*n - 6)*x^5)*P[n - 2, x] - (1 + (n + 2)*x + 2*n*x^2 - (n^2 - 2*n + 2)*x^3 - (3*n^2 - 5*n + 3)*x^4 - (3*n^2 - 7*n + 4)*x^5 - (n^2 - 3*n + 2)*x^6)*D[P[n - 2, x], x] - (2 + (2*n - 1)*x - 2*x^2 - (2*n^2 - 4*n + 2)*x^3 - (2*n^2 - 8*n + 8)*x^4)*P[n - 1, x] + (x + 2*x^2 - (n - 3)*x^3 - (2*n - 4)*x^4 - (n - 2)*x^5)*D[P[n - 1, x], x])/(-x + (n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 + (n^2 + n - 9)*x - 4*x^2 - (n^3 - n^2 - 5*n + 3)*x^3 - (2*n^3 - 8*n^2 + 6*n)*x^4 - (n^3 - 6*n^2 + 11*n - 6)*x^5)*v[i - 2] - (1 + (n + 2)*x + 2*n*x^2 - (n^2 - 2*n + 2)*x^3 - (3*n^2 - 5*n + 3)*x^4 - (3*n^2 - 7*n + 4)*x^5 - (n^2 - 3*n + 2)*x^6)*deriv(v[i - 2], x) - (2 + (2*n - 1)*x - 2*x^2 - (2*n^2 - 4*n + 2)*x^3 - (2*n^2 - 8*n + 8)*x^4)*v[i - 1] + (x + 2*x^2 - (n - 3)*x^3 - (2*n - 4)*x^4 - (n - 2)*x^5)*deriv(v[i - 1], x))/(-x + (n - 2)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from shift_x_plus_one of Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

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.