A089503 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(n - 7\,n x + 6\,n x^{2} + 4\,n^{2} x - 5\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-n x^{2} + 2\,n x^{3} - n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	4														4
3	12	60													72
4	24	336	1008												1368
5	40	1080	8640	20160											29920
6	60	2640	39600	237600	475200										755100
7	84	5460	131040	1441440	7207200	12972960									21758184
8	112	10080	352800	6115200	55036800	242161920	403603200								707280112
9	144	17136	822528	20563200	287884800	2245501440	8982005760	14114580480							25651375488
10	180	27360	1723680	58605120	1172102400	14065228800	98456601600	365695948800	548543923200						1027994161140
11	220	41580	3326400	147470400	3981700800	67688913600	722015078400	4641525504000	16245339264000	23465490048000					45146191347400
12	264	60720	6011280	336631680	11782108800	268632080640	4029481209600	39143531750400	234861190502400	782870635008000	1096018889011200				2157204484374984
13	312	85800	10296000	710424000	31258656000	919004486400	18380089728000	249444074880000	2244996673920000	12721647818880000	40709273020416000	55512645027840000			111457337689612512
14	364	117936	16864848	1405404000	75891816000	2792818828800	71682349939200	1290282298905600	16128528736320000	136196464884480000	735460910376192000	2273242813890048000	3030990418520064000		6193383971188980748
15	420	158340	26601120	2633510880	171178207200	7703019324000	246496618368000	5669422222464000	93545466670656000	1091363777824320000	8730910222594560000	45241989335262720000	135725968005788160000	177487804315261440000	368377504919100489960

Technical details

Node id: n-eb5c2993685be52f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d0e0bf31a1431b92

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:   4;
   3:  12,    60;
   4:  24,   336,    1008;
   5:  40,  1080,    8640,    20160;
   6:  60,  2640,   39600,   237600,     475200;
   7:  84,  5460,  131040,  1441440,    7207200,    12972960;
   8: 112, 10080,  352800,  6115200,   55036800,   242161920,   403603200;
   9: 144, 17136,  822528, 20563200,  287884800,  2245501440,  8982005760,  14114580480;
  10: 180, 27360, 1723680, 58605120, 1172102400, 14065228800, 98456601600, 365695948800, 548543923200;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n+(4*n^2-7*n)*x+(n^3-5*n^2+6*n)*x^2)*P(n-1,x) - (n*x^2+(n^2-2*n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n + (4*n^2 - 7*n)*x + (n^3 - 5*n^2 + 6*n)*x^2)*P[n - 1, x] - (n*x^2 + (n^2 - 2*n)*x^3)*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] = 4; for(n = 3, N, my(i = n); v[i] = ((n + (4*n^2 - 7*n)*x + (n^3 - 5*n^2 + 6*n)*x^2)*v[i - 1] - (n*x^2 + (n^2 - 2*n)*x^3)*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 of numbers used for basis change between certain falling factorials.?
  - 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

[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.