type_b_eulerian → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-64\,x + n + 64\,n x + n^{2} + 16\,n^{2} x - 160\,n^{2} x^{2} - 16\,n^{3} x - 208\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 80\,x^{2} - 4\,n x + 80\,n x^{2} - 4\,n^{2} x + 80\,n^{2} x^{2} - 288\,n^{2} x^{3} + 16\,n^{3} x^{2} - 512\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{2} - 16\,x^{3} - 4\,n x^{2} + 16\,n x^{3} - 4\,n^{2} x^{2} + 32\,n^{2} x^{3} - 64\,n^{2} x^{4} + 32\,n^{3} x^{3} - 128\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-n - 16\,n^{2} x + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,n x + 4\,n^{2} x - 16\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	16							17
3	1	80							81
4	1	288	720						1009
5	1	928	8784						9713
6	1	2864	68976	62464					134305
7	1	8688	447984	1418240					1874913
8	1	26176	2630880	19484672	8864000				31005729
9	1	78656	14567904	211226624	323334400				549207585
10	1	236112	77794848	1996556288	6870508800	1862406144			10807502193
11	1	708496	405974304	17295140864	111748486400	99634728960			229085039025
12	1	2125664	2087259696	141274484736	1546209529600	3026995863552	542455746560		5259025009809
13	1	6377184	10627608240	1107663278080	19213678944000	68921622061056	40012546330624		129266144599185
14	1	19131760	53769382416	8433809873920	221383378912000	1311491796258816	1645882929295360	209045140013056	3396290842867329

Technical details

Node id: n-317a2ef12f8e68fe · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-98b8e10712046853

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: 1,    16;
  3: 1,    80;
  4: 1,   288,      720;
  5: 1,   928,     8784;
  6: 1,  2864,    68976,     62464;
  7: 1,  8688,   447984,   1418240;
  8: 1, 26176,  2630880,  19484672,   8864000;
  9: 1, 78656, 14567904, 211226624, 323334400;

%F Formula

The row polynomials satisfy n^2*P(n,x) = (n^2+n-(16*n^3-16*n^2-64*n+64)*x+(144*n^4-208*n^3-160*n^2)*x^2)*P(n-2,x) - ((4*n^2+4*n-6)*x-(16*n^3+80*n^2+80*n-80)*x^2-(64*n^4-512*n^3-288*n^2)*x^3)*d/dx P(n-2,x) - ((4*n^2+4*n-4)*x^2-(32*n^3+32*n^2+16*n-16)*x^3+(128*n^3+64*n^2)*x^4)*d^2/dx^2 P(n-2,x) - (n-(16*n^3-16*n^2)*x)*P(n-1,x) + ((4*n^2+2*n-2)*x-(16*n^3+16*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^2 + n - (16*n^3 - 16*n^2 - 64*n + 64)*x + (144*n^4 - 208*n^3 - 160*n^2)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n - 6)*x - (16*n^3 + 80*n^2 + 80*n - 80)*x^2 - (64*n^4 - 512*n^3 - 288*n^2)*x^3)*D[P[n - 2, x], x] - ((4*n^2 + 4*n - 4)*x^2 - (32*n^3 + 32*n^2 + 16*n - 16)*x^3 + (128*n^3 + 64*n^2)*x^4)*D[P[n - 2, x], {x, 2}] - (n - (16*n^3 - 16*n^2)*x)*P[n - 1, x] + ((4*n^2 + 2*n - 2)*x - (16*n^3 + 16*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^2 + n - (16*n^3 - 16*n^2 - 64*n + 64)*x + (144*n^4 - 208*n^3 - 160*n^2)*x^2)*v[i - 2] - ((4*n^2 + 4*n - 6)*x - (16*n^3 + 80*n^2 + 80*n - 80)*x^2 - (64*n^4 - 512*n^3 - 288*n^2)*x^3)*deriv(v[i - 2], x) - ((4*n^2 + 4*n - 4)*x^2 - (32*n^3 + 32*n^2 + 16*n - 16)*x^3 + (128*n^3 + 64*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - (16*n^3 - 16*n^2)*x)*v[i - 1] + ((4*n^2 + 2*n - 2)*x - (16*n^3 + 16*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 gamma_polynomial of type_b_eulerian?
  - 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)

Related ?

Same fingerprint ?

A060187 → γ → (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
[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.