A201637 → reverse (shift +1) → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	4	3													7
3	0	12	24	4												40
4	0	48	174	88	5											315
5	0	240	1332	1312	260	6										3150
6	0	1440	11124	17600	7260	684	7									38115
7	0	10080	101952	232560	160600	33660	1680	8								540540
8	0	80640	1023408	3141216	3220100	1174800	139650	3952	9							8783775
9	0	725760	11201760	44105184	62200320	34593000	7437500	538080	9036	10						160810650
10	0	7257600	133017120	648747648	1194524520	932144304	313355000	42609280	1969632	20260	11					3273645375
11	0	79833600	1705069440	10029924864	23210819760	24016170528	11538690128	2514957760	227195136	6950380	44792	12				73329656400
12	0	958003200	23483157120	163153207296	460998951120	607082598144	392289791488	124391686272	18463344984	1148763760	23876886	97992	13			1791993478275
13	0	12454041600	346008257280	2791717747200	9415398309120	15299192840544	12727320496128	5501760371072	1208910782016	126698177760	5579852036	80396352	212628	14		47435121483750
14	0	174356582400	5432977670400	50203616071680	198447890279040	388370250839232	402460614252144	226399280528896	68713489344528	10846765124160	825176574508	26277957696	266603844	458332	15	1351900962286875

Technical details

Node id: n-b818d5ca6ab31711 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e5cc694a96139667

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: 0,      2;
  2: 0,      4,        3;
  3: 0,     12,       24,        4;
  4: 0,     48,      174,       88,        5;
  5: 0,    240,     1332,     1312,      260,        6;
  6: 0,   1440,    11124,    17600,     7260,      684,       7;
  7: 0,  10080,   101952,   232560,   160600,    33660,    1680,      8;
  8: 0,  80640,  1023408,  3141216,  3220100,  1174800,  139650,   3952,    9;
  9: 0, 725760, 11201760, 44105184, 62200320, 34593000, 7437500, 538080, 9036, 10;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + n + 6 + (2*n^3 - 6*n^2 - 2*n + 6)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n^2 - 2*n - 3)*x + (3*n + 3)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 3*n - 3 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + n + 6 + (2*n^3 - 6*n^2 - 2*n + 6)*x + (n^3 - n)*x^2)*v[i - 2] - ((n^2 - 2*n - 3)*x + (3*n + 3)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 3*n - 3 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); 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 Row reversed version of triangle A201637 (second-order Eulerian triangle).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    ↑ Liu–Wang theorem (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.