A330798 → D

nonnegative coefficients

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

Recurrence

\(\left(-2\,n^{2} - 2\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(2\,n - 2\,n^{2} - 6\,n^{2} x + 10\,n^{3} + 12\,n^{3} x + 14\,n^{4} + 18\,n^{4} x\right)\,P_{n-1}(x) + \left(1 + x - n - 4\,n x - 3\,n x^{2} - n^{2} + 5\,n^{2} x + 6\,n^{2} x^{2} + n^{3} + 10\,n^{3} x + 9\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	15	18													33
3	84	224	144												452
4	420	1800	2475	1100											5795
5	1980	11880	25740	24024	8190										71814
6	9009	70070	210210	305760	216580	59976									871605
7	40040	384384	1485120	2970240	3255840	1860480	434112								10430216
8	175032	2004912	9523332	24418800	36628200	32232816	15444891	3114936							123542919
9	755820	10077600	56977200	179071200	343219800	411863760	302841000	124982000	22200750						1451989130
10	3233230	49244580	323607240	1208133696	2831563350	4330626300	4330626300	2735132400	991485495	157378650					16961031241
11	13728792	235350720	1765130400	7648898400	21259438200	39684284640	50475625200	43264821600	23950169100	7741468800	1110732480				197149648332
12	57946200	1104840880	9322094925	46062116100	148422374100	328091563800	508541923890	553515019200	415136264400	204559898400	59663303700	7810468848			2282287814443
13	243374040	5110854840	47951843940	266399133000	977965238250	2503591009920	4589916851520	6080149595520	5782750974000	3855167316000	1711694288304	454855824864	54751164104		26330547468302
14	1017958725	23353170750	241316097750	1490232744000	6147210069000	17914726486800	38000934972000	59479724304000	68773431226500	58075341924600	34845205154760	14078870769600	3435914890200	382780969200	302890060737885

Technical details

Node id: n-9b6812a2cbd6bf83 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-c25c6caf35317535

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 -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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      0;
  1:      2;
  2:     15,       18;
  3:     84,      224,      144;
  4:    420,     1800,     2475,      1100;
  5:   1980,    11880,    25740,     24024,      8190;
  6:   9009,    70070,   210210,    305760,    216580,     59976;
  7:  40040,   384384,  1485120,   2970240,   3255840,   1860480,    434112;
  8: 175032,  2004912,  9523332,  24418800,  36628200,  32232816,  15444891,   3114936;
  9: 755820, 10077600, 56977200, 179071200, 343219800, 411863760, 302841000, 124982000, 22200750;

%F Formula

The row polynomials satisfy (4*n^4-2*n^3-2*n^2)*P(n,x) = (14*n^4+10*n^3-2*n^2+2*n+(18*n^4+12*n^3-6*n^2)*x)*P(n-1,x) + (n^3-n^2-n+1+(10*n^3+5*n^2-4*n+1)*x+(9*n^3+6*n^2-3*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^4 + 10*n^3 - 2*n^2 + 2*n + (18*n^4 + 12*n^3 - 6*n^2)*x)*P[n - 1, x] + (n^3 - n^2 - n + 1 + (10*n^3 + 5*n^2 - 4*n + 1)*x + (9*n^3 + 6*n^2 - 3*n)*x^2)*D[P[n - 1, x], x])/(4*n^4 - 2*n^3 - 2*n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((14*n^4 + 10*n^3 - 2*n^2 + 2*n + (18*n^4 + 12*n^3 - 6*n^2)*x)*v[i - 1] + (n^3 - n^2 - n + 1 + (10*n^3 + 5*n^2 - 4*n + 1)*x + (9*n^3 + 6*n^2 - 3*n)*x^2)*deriv(v[i - 1], x))/(4*n^4 - 2*n^3 - 2*n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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, interpolating between the central binomial coefficients and the central coefficients of the Cata…?
  - 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 ?

A104978 → D

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.