A225468 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 3\,x^{2}\right)\,P_{n-2}'(x) + \left(7 + 2\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

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

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	7	2													9
4	39	30	3												72
5	203	318	78	4											603
6	1031	2950	1335	160	5										5481
7	5187	25662	19110	4020	285	6									54270
8	25999	215670	248703	81760	9870	462	7								582471
9	130123	1777358	3059238	1476244	269990	21084	700	8							6734745
10	650871	14479110	36317655	24746400	6435135	745668	40698	1008	9						83416554
11	3254867	117134622	421212870	394873140	140330295	22635522	1806000	72720	1395	10					1101321441
12	16275359	943586710	4809043503	6089841120	2879206440	621106794	67946109	3954720	122265	1870	11				15431084901
13	81378843	7581244398	54314858598	91669833684	56558810880	15877183608	2287385100	180475416	7994745	195690	2442	12			228559363416
14	406898311	60812712870	608835311175	1355797483040	1076087077065	385414245216	71133238176	7306515216	434882448	15145130	300729	3120	13		3566243812509
15	2034499747	487315499582	6788407492230	19792945177460	19988227163905	8999589396798	2085714430800	271264524960	20831420610	967846880	27185158	446628	3913	14	58437325088685

Technical details

Node id: n-da83f3909ce4eab5 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-bfeb907a8d7b5f35

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:      7,        2;
   4:     39,       30,        3;
   5:    203,      318,       78,        4;
   6:   1031,     2950,     1335,      160,       5;
   7:   5187,    25662,    19110,     4020,     285,      6;
   8:  25999,   215670,   248703,    81760,    9870,    462,     7;
   9: 130123,  1777358,  3059238,  1476244,  269990,  21084,   700,    8;
  10: 650871, 14479110, 36317655, 24746400, 6435135, 745668, 40698, 1008, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(10+4*x+x^2)*P(n-2,x) - (6*x+3*x^2)*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + 3*x*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[-(10 + 4*x + x^2)*P[n - 2, x] - (6*x + 3*x^2)*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
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] = -(10 + 4*x + x^2)*v[i - 2] - (6*x + 3*x^2)*deriv(v[i - 2], x) + (7 + 2*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[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.