type_b_eulerian → γ (scale 1/4) → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(16\,x + n + n^{2} - 4\,n^{2} x - 10\,n^{2} x^{2} - n^{3} + 8\,n^{3} x - 13\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 20\,x^{2} - 2\,n^{2} x - 20\,n^{2} x^{2} - 18\,n^{2} x^{3} + 4\,n^{3} x - 32\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{2} + 4\,x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3} - 4\,n^{2} x^{4} - 8\,n^{3} x^{3} - 8\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(n + 2\,n^{2} - 4\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 4\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	2	4			6
3	6	20			26
4	24	112	45		181
5	120	720	549		1389
6	720	5280	5958	976	12934
7	5040	43680	65394	22160	136274

All 15 rows

Technical details

Node id: n-b47eb1d248b2ecb3 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-afd5149745c5a1a0

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:      2,       4;
  3:      6,      20;
  4:     24,     112,      45;
  5:    120,     720,     549;
  6:    720,    5280,    5958,     976;
  7:   5040,   43680,   65394,   22160;
  8:  40320,  403200,  752976,  393088,   34625;
  9: 362880, 4112640, 9208080, 6533760, 1263025;

%F Formula

The row polynomials satisfy n^2*P(n,x) = -(n^4+n^3-n^2-n+(4*n^4-8*n^3+4*n^2-16)*x-(9*n^4-13*n^3-10*n^2)*x^2)*P(n-2,x) + ((4*n^3-2*n^2+6)*x+(4*n^4-20*n^2+20)*x^2+(4*n^4-32*n^3-18*n^2)*x^3)*d/dx P(n-2,x) - ((4*n^2-4)*x^2+(8*n^3+8*n^2-4)*x^3+(8*n^3+4*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3+2*n^2+n+(4*n^3-4*n^2)*x)*P(n-1,x) - ((4*n^2-2)*x+(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + n^3 - n^2 - n + (4*n^4 - 8*n^3 + 4*n^2 - 16)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*P[n - 2, x] + ((4*n^3 - 2*n^2 + 6)*x + (4*n^4 - 20*n^2 + 20)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4)*x^2 + (8*n^3 + 8*n^2 - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 + 2*n^2 + n + (4*n^3 - 4*n^2)*x)*P[n - 1, x] - ((4*n^2 - 2)*x + (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + n^3 - n^2 - n + (4*n^4 - 8*n^3 + 4*n^2 - 16)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*v[i - 2] + ((4*n^3 - 2*n^2 + 6)*x + (4*n^4 - 20*n^2 + 20)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4)*x^2 + (8*n^3 + 8*n^2 - 4)*x^3 + (8*n^3 + 4*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 + 2*n^2 + n + (4*n^3 - 4*n^2)*x)*v[i - 1] - ((4*n^2 - 2)*x + (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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 shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.?
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from type_b_eulerian?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from type_b_eulerian?
    
    ↑ Liu–Wang theorem (above)
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from type_b_eulerian?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.?
    
    ↑ Gamma criterion (above)
    
    ↑ Gamma criterion (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.