A059344 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(24 + 63\,x + 36\,n x + 27\,n x^{2} + 27\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-24 - 279\,x - 756\,x^{2} - 324\,x^{3} + 48\,n + 270\,n x + 297\,n x^{2} - 162\,n x^{3} + 108\,n^{2} x + 459\,n^{2} x^{2} + 324\,n^{2} x^{3} + 54\,n^{3} x^{2} + 162\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-16 - 180\,x - 720\,x^{2} - 1188\,x^{3} - 648\,x^{4} - 48\,n x - 396\,n x^{2} - 1080\,n x^{3} - 972\,n x^{4} - 36\,n^{2} x^{2} - 216\,n^{2} x^{3} - 324\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	3	12			15
3	7	36			43
4	25	216	324		565
5	81	840	1620		2541
6	331	4500	14580	12960	32371
7	1303	20412	79380	90720	191815

All 15 rows

Technical details

Node id: n-b92b7c60e520404c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1006d0233b977996

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     1;
  2:     3,     12;
  3:     7,     36;
  4:    25,    216,     324;
  5:    81,    840,    1620;
  6:   331,   4500,   14580,   12960;
  7:  1303,  20412,   79380,   90720;
  8:  5937, 111216,  567000, 1088640,  680400;
  9: 26785, 562896, 3306744, 7620480, 6123600;

%F Formula

The row polynomials satisfy (24+(36*n+63)*x+(27*n^2+27*n)*x^2)*P(n,x) = (48*n-24+(108*n^2+270*n-279)*x+(54*n^3+459*n^2+297*n-756)*x^2+(162*n^3+324*n^2-162*n-324)*x^3)*P(n-1,x) - (16+(48*n+180)*x+(36*n^2+396*n+720)*x^2+(216*n^2+1080*n+1188)*x^3+(324*n^2+972*n+648)*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[((48*n - 24 + (108*n^2 + 270*n - 279)*x + (54*n^3 + 459*n^2 + 297*n - 756)*x^2 + (162*n^3 + 324*n^2 - 162*n - 324)*x^3)*P[n - 1, x] - (16 + (48*n + 180)*x + (36*n^2 + 396*n + 720)*x^2 + (216*n^2 + 1080*n + 1188)*x^3 + (324*n^2 + 972*n + 648)*x^4)*D[P[n - 1, x], x])/(24 + (36*n + 63)*x + (27*n^2 + 27*n)*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] = ((48*n - 24 + (108*n^2 + 270*n - 279)*x + (54*n^3 + 459*n^2 + 297*n - 756)*x^2 + (162*n^3 + 324*n^2 - 162*n - 324)*x^3)*v[i - 1] - (16 + (48*n + 180)*x + (36*n^2 + 396*n + 720)*x^2 + (216*n^2 + 1080*n + 1188)*x^3 + (324*n^2 + 972*n + 648)*x^4)*deriv(v[i - 1], x))/(24 + (36*n + 63)*x + (27*n^2 + 27*n)*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 basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in t…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Parent-row provers of polytool 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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

References

[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.