A286724 → D

nonnegative coefficients

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

Recurrence

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

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

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

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	8	2													10
4	72	36	3												111
5	768	576	96	4											1444
6	9600	9600	2400	200	5										21805
7	138240	172800	57600	7200	360	6									376206
8	2257920	3386880	1411200	235200	17640	588	7								7309435
9	41287680	72253440	36126720	7526400	752640	37632	896	8							157985416
10	836075520	1672151040	975421440	243855360	30481920	2032128	72576	1296	9						3760091289
11	18579456000	41803776000	27869184000	8128512000	1219276800	101606400	4838400	129600	1800	10					97706781010
12	449622835200	1124057088000	843042816000	281014272000	49177497600	4917749760	292723200	10454400	217800	2420	11				2752135656391
13	11771943321600	32372844134400	26977370112000	10116513792000	2023302758400	236051988480	16860856320	752716800	20908800	348480	3168	12			83515660940460
14	331576403558400	994729210675200	911835109785600	379931295744000	85484541542400	11397938872320	949828239360	50883655680	1766793600	39262080	535392	4056	13		2715957018668101
15	9998303861145600	32494487548723200	32494487548723200	14893306793164800	3723326698291200	558499004743680	53190381404160	3324398837760	138516618240	3847683840	69957888	794976	5096	14	94219068670093654

Technical details

Node id: n-34454f359e0dcb07 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-41ba2c3c941da546

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:         8,          2;
   4:        72,         36,         3;
   5:       768,        576,        96,         4;
   6:      9600,       9600,      2400,       200,        5;
   7:    138240,     172800,     57600,      7200,      360,       6;
   8:   2257920,    3386880,   1411200,    235200,    17640,     588,     7;
   9:  41287680,   72253440,  36126720,   7526400,   752640,   37632,   896,    8;
  10: 836075520, 1672151040, 975421440, 243855360, 30481920, 2032128, 72576, 1296, 9;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n^2-4*n+2+(n-1)*x)*P(n-1,x) + (2*n-2)*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[Together[((2*n^2 - 4*n + 2 + (n - 1)*x)*P[n - 1, x] + (2*n - 2)*x*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 - 4*n + 2 + (n - 1)*x)*v[i - 1] + (2*n - 2)*x*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. A generalization of unsigned Lah numbers, called L[2,1].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.