A112493 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - 2\,x + 2\,x^{2} - 2\,n + 4\,n x - 5\,n x^{2} - 2\,n^{2} x + 4\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(x - x^{2} - 2\,x^{3} - n x + 2\,n x^{2} + 3\,n x^{3} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-5 + 3\,n - 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2} + n x + n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	4	6													10
4	11	50	45												106
5	26	260	630	420											1336
6	57	1092	5250	8820	4725										19944
7	120	4074	34104	107100	138600	62370									346368
8	247	14142	191415	990220	2304225	2432430	945945								6878624
9	502	46872	977790	7758520	28678650	52972920	47297250	16216200							153948704
10	1013	150656	4684548	54437240	298147850	852611760	1305404100	1010809800	310134825						3836381792
11	2036	474254	21450000	353385032	2741538800	11334943620	26358652320	34459425000	23570246700	6547290750					105387408512
12	4083	1471626	95047953	2167325160	23075202150	131982150300	435678933690	852369958440	972272676375	595803458250	151242416325				3164688644352
13	8178	4521036	411095958	12737584860	181797716100	1394609175960	6257056625820	17131786471800	28901050828650	29240200489500	16233352685550	3794809718700			103147816922112
14	16369	13792092	1746107142	72457971404	1361459974875	13689094899480	80970289740340	297204515407800	693748963682775	1028112336751500	934577304610950	474351214837500	102776096548125		3626865494340352
15	32752	41867346	7315438776	401868285572	9799064205840	126864951723510	967174532243920	4618127592438360	14293301393170800	29038375887270750	38357093345571000	31655037736822500	14799757902930000	2988412653476250	136854354285477376

Technical details

Node id: n-ec648b55f50a233f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-b73aa855d7a17944

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    0;
   2:    1;
   3:    4,      6;
   4:   11,     50,      45;
   5:   26,    260,     630,      420;
   6:   57,   1092,    5250,     8820,      4725;
   7:  120,   4074,   34104,   107100,    138600,     62370;
   8:  247,  14142,  191415,   990220,   2304225,   2432430,     945945;
   9:  502,  46872,  977790,  7758520,  28678650,  52972920,   47297250,   16216200;
  10: 1013, 150656, 4684548, 54437240, 298147850, 852611760, 1305404100, 1010809800, 310134825;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(2*n-2+(2*n^2-4*n+2)*x+(n^3-4*n^2+5*n-2)*x^2)*P(n-2,x) - ((n-1)*x+(n^2-2*n+1)*x^2+(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (3*n-5+(2*n^2-4*n)*x)*P(n-1,x) + ((n-2)*x+(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 + (2*n^2 - 4*n + 2)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*P[n - 2, x] - ((n - 1)*x + (n^2 - 2*n + 1)*x^2 + (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (3*n - 5 + (2*n^2 - 4*n)*x)*P[n - 1, x] + ((n - 2)*x + (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n - 2 + (2*n^2 - 4*n + 2)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*v[i - 2] - ((n - 1)*x + (n^2 - 2*n + 1)*x^2 + (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (3*n - 5 + (2*n^2 - 4*n)*x)*v[i - 1] + ((n - 2)*x + (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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.