basis-map coefficients of A297191, shifted → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 1 + 6\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	6	3													10
2	1	30	75	40	5											151
3	1	126	903	1400	700	126	7									3263
4	1	510	9075	31080	34755	15876	3234	288	9							94828
5	1	2046	85503	583000	1233650	1076922	447909	95040	10395	550	11					3535027
6	1	8190	784875	10130120	37542505	55927872	40007968	15196896	3235518	393250	26741	936	13			163254885
7	1	32766	7125303	169423800	1053834600	2524159638	2861189331	1733022720	604156410	126626500	16274258	1277640	59150	1470	15	9097183602

All 15 rows

Technical details

Node id: n-5b410a2e2cae6c98 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-39851ca50dcc7dd5

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,      6,         3;
  2: 1,     30,        75,          40,            5;
  3: 1,    126,       903,        1400,          700,           126,              7;
  4: 1,    510,      9075,       31080,        34755,         15876,           3234,            288,              9;
  5: 1,   2046,     85503,      583000,      1233650,       1076922,         447909,          95040,          10395,           550,            11;
  6: 1,   8190,    784875,    10130120,     37542505,      55927872,       40007968,       15196896,        3235518,        393250,         26741,          936,          13;
  7: 1,  32766,   7125303,   169423800,   1053834600,    2524159638,     2861189331,     1733022720,      604156410,     126626500,      16274258,      1277640,       59150,       1470,        15;
  8: 1, 131070,  64373475,  2777349160,  28263758255,  105034499388,   179956733502,   163327960224,    85759400727,   27583341500,    5632670758,    744267888,    63832314,    3498600,    117300,    2176,     17;
  9: 1, 524286, 580344303, 45038667800, 737946423550, 4158489610674, 10450472443873, 13678008027840, 10301531641245, 4772970337850, 1423545395701, 281603415600, 37601709080, 3410085420, 208751670, 8434176, 214149, 3078, 19;

%F Formula

The row polynomials satisfy P(n,x) = -(2*x^2+2*x^3+x^4)*P(n-2,x) - (2*x^2+3*x^3+2*x^4)*d/dx P(n-2,x) - (x^3+x^4)*d^2/dx^2 P(n-2,x) + (1+6*x+2*x^2)*P(n-1,x) + (3*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 6*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(2*x^2 + 2*x^3 + x^4)*P[n - 2, x] - (2*x^2 + 3*x^3 + 2*x^4)*D[P[n - 2, x], x] - (x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (1 + 6*x + 2*x^2)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {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; v[2] = 1 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(2*x^2 + 2*x^3 + x^4)*v[i - 2] - (2*x^2 + 3*x^3 + 2*x^4)*deriv(v[i - 2], x) - (x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 6*x + 2*x^2)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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_A008277 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the …?
  - Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
    
    from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    
    Parent-row provers of polytool polytool ? 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 basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.