A151775 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 24\,x - 24\,x^{2}\right)\,P_{n-2}(x) + \left(-6 - 32\,x - 62\,x^{2} - 36\,x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x - 16\,x^{2} - 20\,x^{3} - 8\,x^{4}\right)\,P_{n-2}''(x) + \left(14 + 24\,x\right)\,P_{n-1}(x) + \left(6 + 28\,x + 22\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x + 8\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

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

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	28	48													76
3	662	2640	2160												5462
4	24568	167328	302400	161280											655576
5	1326122	13498080	40007520	45964800	18144000										118940522
6	98329108	1385398608	5938574400	10958492160	9180864000	2874009600									30435667876
7	9596075582	177932729520	1027378432080	2693417126400	3577942368000	2353813862400	610248038400								10450328632382
8	1192744081648	28073116365888	208457658864000	724868256752640	1352654214912000	1398688504012800	756707567616000	167382319104000							4638024381708976
9	183983154281042	5348776806240960	49426658023515840	218061839231308800	534339995821632000	770096309946163200	650712365330227200	298777439600640000	57621363351552000						2584568731265561042
10	34489251602450188	1212248605228941168	13600558135994659200	73781970186786462720	227331690084180864000	427190729342811033600	500210284598728704000	356987821333118976000	142324767478333440000	24329020081766400000					1767003579098551930876
11	7722592644581974502	322541097572738576400	4309441376639879818800	28081775703965993856000	105531755382326661696000	247752842391035780198400	376262803826476645478400	370471234584879267840000	228706048084548722688000	80553385490728550400000	12364008005553684480000				1454363558536372507006502
12	2035778987564783402728	99598354756749845396448	1560251445971734762488000	11991915487391699777702400	53734530369204302365440000	152959867117885099109376000	288811786541467911321600000	367318379834447190589440000	311747623281589727232000000	169503056417955830169600000	53461970616014131691520000	7445380820798873272320000			1418636396066470814920285576
13	624086284447168370787962	35337420160162993712679840	640763749342102588517820960	5724212706887062433677632000	30050007086893077949785120000	101442626718848991682089984000	231205100350556170067733504000	364200817708025894640844800000	397966519500838302432337920000	296632872380675784425472000000	144065792200951698171494400000	41135729034913774829568000000	5242788994645873262592000000		1618343191939023342646196648762
14	220145816803809737344107268	14279087284235466553475203728	296472036290542676955545918400	3041455046039135991877115911680	18443714434801553835721143360000	72579458940250360474306554624000	195359170039571854706923653120000	370152595051254680345148948480000	499431639708253269669773721600000	478006186630235331183058944000000	317502152135089873956510105600000	139303484653092062863137177600000	36332527732895901709762560000000	4268436824563994047021056000000	2134731792465439600736487345525076

Technical details

Node id: n-789f42810b77fd52 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3764fa9cd037f7ff

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:              28,               48;
  3:             662,             2640,              2160;
  4:           24568,           167328,            302400,             161280;
  5:         1326122,         13498080,          40007520,           45964800,           18144000;
  6:        98329108,       1385398608,        5938574400,        10958492160,         9180864000,         2874009600;
  7:      9596075582,     177932729520,     1027378432080,      2693417126400,      3577942368000,      2353813862400,       610248038400;
  8:   1192744081648,   28073116365888,   208457658864000,    724868256752640,   1352654214912000,   1398688504012800,    756707567616000,    167382319104000;
  9: 183983154281042, 5348776806240960, 49426658023515840, 218061839231308800, 534339995821632000, 770096309946163200, 650712365330227200, 298777439600640000, 57621363351552000;

%F Formula

The row polynomials satisfy P(n,x) = -(9+24*x+24*x^2)*P(n-2,x) - (6+32*x+62*x^2+36*x^3)*d/dx P(n-2,x) - (4*x+16*x^2+20*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (14+24*x)*P(n-1,x) + (6+28*x+22*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^2 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[-(9 + 24*x + 24*x^2)*P[n - 2, x] - (6 + 32*x + 62*x^2 + 36*x^3)*D[P[n - 2, x], x] - (4*x + 16*x^2 + 20*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (14 + 24*x)*P[n - 1, x] + (6 + 28*x + 22*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {x, 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] = -(9 + 24*x + 24*x^2)*v[i - 2] - (6 + 32*x + 62*x^2 + 36*x^3)*deriv(v[i - 2], x) - (4*x + 16*x^2 + 20*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (14 + 24*x)*v[i - 1] + (6 + 28*x + 22*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x)); 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: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.