A292219 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4 - 3\,x + n\right)\,P_{n}(x) = \left(-10 + 7\,x - 3\,x^{2} + 18\,n - 11\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(4\,x - 12\,x^{2} + 4\,n x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	7	1														8
3	80	22	1													103
4	1260	504	45	1												1810
5	25200	13104	1728	76	1											40109
6	609840	388080	67320	4400	115	1										1069756
7	17297280	12972960	2779920	240240	9360	162	1									33299923
8	562161600	484323840	123243120	13213200	682500	17640	217	1								1183642118
9	20583763200	19995655680	5881075200	751470720	48266400	1656480	30464	280	1							47261918425
10	838053216000	905097473280	301699157760	44696171520	3428399520	146512800	3581424	49248	351	1						2093124561904
11	37544784076800	44584431091200	16593453676800	2793510720000	248876409600	12717311040	387444960	7086240	75600	430	1					101778167892671
12	1835001321753600	2374707592857600	975326332780800	183757135161600	18632716502400	1107089786880	40362648480	920636640	13069980	111320	517	1				5388573485309818
13	97147128798720000	136005980318208000	61063909530624000	12721647818880000	1445641797600000	97858829376000	4151586700800	113383670400	2008314000	22770000	158400	612	1			308486434095022213
14	5537386341527040000	8335223650930176000	4059362167660800000	926135961214464000	116518728886560000	8847327801312000	427909972089600	13606040448000	288680792400	4084938000	37837800	219024	715	1		18983915986836677540
15	338072008219545600000	544295933233468416000	285670846945516032000	70807303943760384000	9766524681897984000	821702797755840000	44606723306745600	1617758209267200	39976695158400	677815538400	7837830000	60422544	295568	826	1	1249480584965869514539

Technical details

Node id: n-f8039dbc976d8dfa · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a38c1fc9c9cd2b68

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 -1)
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:            1;
   2:            7,            1;
   3:           80,           22,            1;
   4:         1260,          504,           45,           1;
   5:        25200,        13104,         1728,          76,          1;
   6:       609840,       388080,        67320,        4400,        115,         1;
   7:     17297280,     12972960,      2779920,      240240,       9360,       162,       1;
   8:    562161600,    484323840,    123243120,    13213200,     682500,     17640,     217,     1;
   9:  20583763200,  19995655680,   5881075200,   751470720,   48266400,   1656480,   30464,   280,   1;
  10: 838053216000, 905097473280, 301699157760, 44696171520, 3428399520, 146512800, 3581424, 49248, 351, 1;

%F Formula

The row polynomials satisfy (n+4-3*x)*P(n,x) = (4*n^2+18*n-10-(11*n-7)*x-3*x^2)*P(n-1,x) + ((4*n+4)*x-12*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 18*n - 10 - (11*n - 7)*x - 3*x^2)*P[n - 1, x] + ((4*n + 4)*x - 12*x^2)*D[P[n - 1, x], x])/(n + 4 - 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^2 + 18*n - 10 - (11*n - 7)*x - 3*x^2)*v[i - 1] + ((4*n + 4)*x - 12*x^2)*deriv(v[i - 1], x))/(n + 4 - 3*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

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[4,3].?
  - 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

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.