A172431 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8\,n - 20\,n^{3} x + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(8 + 16\,x - 4\,x^{2} + 8\,n + 16\,n x - 22\,n x^{2} - 20\,n^{2} x - 36\,n^{2} x^{2} + n^{2} x^{3} - 20\,n^{3} x - 18\,n^{3} x^{2} + 3\,n^{3} x^{3} + 2\,n^{4} x^{2} + 3\,n^{4} x^{3} + 2\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 4\,x^{2} - 2\,x^{3} - 40\,n x^{2} - 10\,n x^{3} - 40\,n^{2} x^{2} - 6\,n^{2} x^{3} + n^{2} x^{4} + 8\,n^{3} x^{3} + 2\,n^{3} x^{4} + 4\,n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	3	16	9						28
3	4	40	54	16					114
4	5	80	189	128	25				427
5	6	140	504	576	250	36			1512
6	7	224	1134	1920	1375	432	49		5141
7	8	336	2268	5280	5500	2808	686	64	16950

All 15 rows

Technical details

Node id: n-c38fb046add480e8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-b5575767b5ed4a03

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:  2,   4;
  2:  3,  16,    9;
  3:  4,  40,   54,    16;
  4:  5,  80,  189,   128,    25;
  5:  6, 140,  504,   576,   250,    36;
  6:  7, 224, 1134,  1920,  1375,   432,    49;
  7:  8, 336, 2268,  5280,  5500,  2808,   686,   64;
  8:  9, 480, 4158, 12672, 17875, 13104,  5145, 1024,   81;
  9: 10, 660, 7128, 27456, 50050, 49140, 27440, 8704, 1458, 100;

%F Formula

The row polynomials satisfy (8*n-20*n^3*x+2*n^5*x^2)*P(n,x) = (8*n+8-(20*n^3+20*n^2-16*n-16)*x+(2*n^5+2*n^4-18*n^3-36*n^2-22*n-4)*x^2+(n^5+3*n^4+3*n^3+n^2)*x^3)*P(n-1,x) + (16*x-(40*n^2+40*n+4)*x^2+(4*n^4+8*n^3-6*n^2-10*n-2)*x^3+(n^4+2*n^3+n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 8 - (20*n^3 + 20*n^2 - 16*n - 16)*x + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)*P[n - 1, x] + (16*x - (40*n^2 + 40*n + 4)*x^2 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 20*n^3*x + 2*n^5*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 8 - (20*n^3 + 20*n^2 - 16*n - 16)*x + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)*v[i - 1] + (16*x - (40*n^2 + 40*n + 4)*x^2 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 20*n^3*x + 2*n^5*x^2)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 reversal of Even row Pascal-square read by antidiagonals.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Even row Pascal-square read by antidiagonals.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

A078812 → (θ+1)²

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023