A371259 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-400 - 32\,x - x^{2} - 360\,n - 16\,n x - 121\,n^{2} - 2\,n^{2} x - 18\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(-176\,x - 10\,x^{2} - 120\,n x - 2\,n x^{2} - 27\,n^{2} x - 2\,n^{3} x\right)\,P_{n-2}'(x) + \left(-16\,x^{2} - x^{3} - 8\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(52 + 2\,x + 20\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(13\,x + 2\,n x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	100	2													102
3	9864	400	3												10267
4	1099296	68128	1032	4											1168460
5	142159392	12009024	276600	2160	5										154447181
6	21385410048	2313843840	72227136	854800	3980	6									23772339810
7	3724783667712	496284212736	19721871360	315131648	2218720	6720	7								4241107108903
8	745991108093952	119113515201024	5793225387264	117368874752	1112779840	5088384	10640	8							871016335435864
9	170540762379724800	31985042107650048	1852912409720832	45751815936000	548424615680	3370689408	10628688	16032	9						204425020518981497
10	44186216728692326400	9584758693870313472	648321183911024640	18986955432523776	276559616192000	2144583567744	9076755968	20628000	23220	10					54438562275203355230
11	12888567485475707289600	3193834250271366217728	248421085432685365248	8459210418224209920	145696485031326720	1369849986218496	7303618069440	22255163392	37719000	32560	11				16339429105258911612115
12	4206296083120337151590400	1178751299318914619179008	104159185548481195425792	4061739908347123728384	81091111777942394880	899486424747176448	5805042325252352	22256469199360	50535102816	65646240	44440	12			5493350304321632174740132
13	1527258548958215820764774400	479913111893806521974784000	47702327775858579111542784	2104761029704649478438912	47974373015965431552000	615413514787904507904	4677552282154425856	21679707264770048	61878617802144	107616766400	109584904	59280	13		2057027344143410300429008645
14	613749974379661139635077120000	214696199443085107832409292800	23807796311356262072948883456	1177021688414904346242908160	30264190144326900422000640	442202694361598835302400	3880007776655613804544	21150280632765280256	72902507029747200	159259720322560	217004856640	176515248	77532	14	853461702116587221658246111450

Technical details

Node id: n-21362746ac4a5e84 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-1b006de61aa7a72b

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:                  1;
  2:                100,                 2;
  3:               9864,               400,                3;
  4:            1099296,             68128,             1032,              4;
  5:          142159392,          12009024,           276600,           2160,            5;
  6:        21385410048,        2313843840,         72227136,         854800,         3980,          6;
  7:      3724783667712,      496284212736,      19721871360,      315131648,      2218720,       6720,        7;
  8:    745991108093952,   119113515201024,    5793225387264,   117368874752,   1112779840,    5088384,    10640,     8;
  9: 170540762379724800, 31985042107650048, 1852912409720832, 45751815936000, 548424615680, 3370689408, 10628688, 16032, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4+18*n^3+121*n^2+360*n+400+(2*n^2+16*n+32)*x+x^2)*P(n-2,x) - ((2*n^3+27*n^2+120*n+176)*x+(2*n+10)*x^2)*d/dx P(n-2,x) - ((n^2+8*n+16)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+20*n+52+2*x)*P(n-1,x) + (2*n+13)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^4 + 18*n^3 + 121*n^2 + 360*n + 400 + (2*n^2 + 16*n + 32)*x + x^2)*P[n - 2, x] - ((2*n^3 + 27*n^2 + 120*n + 176)*x + (2*n + 10)*x^2)*D[P[n - 2, x], x] - ((n^2 + 8*n + 16)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 20*n + 52 + 2*x)*P[n - 1, x] + (2*n + 13)*x*D[P[n - 1, x], x] + x^2*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 18*n^3 + 121*n^2 + 360*n + 400 + (2*n^2 + 16*n + 32)*x + x^2)*v[i - 2] - ((2*n^3 + 27*n^2 + 120*n + 176)*x + (2*n + 10)*x^2)*deriv(v[i - 2], x) - ((n^2 + 8*n + 16)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 20*n + 52 + 2*x)*v[i - 1] + (2*n + 13)*x*deriv(v[i - 1], x) + x^2*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, (2, 3)-Lah numbers.?
  - 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.