A225476 → D

nonnegative coefficients

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

Recurrence

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

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

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

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	4	4													8
3	13	36	18												67
4	40	232	288	96											656
5	121	1320	3060	2400	600										7501
6	364	7084	27360	37920	21600	4320									98648
7	1093	36876	223398	487200	474600	211680	35280								1470127
8	3280	188752	1729728	5576256	8265600	6168960	2257920	322560							24513056
9	9841	956880	12957480	59411520	125773200	139708800	84248640	26127360	3265920						452459641
10	29524	4823764	95008320	603810240	1754827200	2721781440	2404684800	1214438400	326592000	36288000					9162283688
11	88573	24236916	686765178	5941003200	23076913200	48017914560	58298153760	42631142400	18501436800	4390848000	439084800				202007587387
12	265720	121538872	4916422368	57131776416	290977315200	790386664320	1266618608640	1257741112320	783167616000	297779328000	63228211200	5748019200			4817816878256
13	797161	608757240	34961881500	540406907040	3557824069800	12370063305600	25454103554880	32962111902720	27614813466240	14955228288000	5057378726400	971415244800	80951270400		123599868171781
14	2391484	3046974844	247472578080	5050125531360	42513607936800	186427156275360	482833736985600	793079134848000	858422838873600	620891054784000	297277972992000	90491066265600	15866448998400	1220496076800	3394324161511928

Technical details

Node id: n-756e6d5ca926111c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-151e0e04582651e2

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:    4,      4;
  3:   13,     36,       18;
  4:   40,    232,      288,       96;
  5:  121,   1320,     3060,     2400,       600;
  6:  364,   7084,    27360,    37920,     21600,      4320;
  7: 1093,  36876,   223398,   487200,    474600,    211680,    35280;
  8: 3280, 188752,  1729728,  5576256,   8265600,   6168960,  2257920,   322560;
  9: 9841, 956880, 12957480, 59411520, 125773200, 139708800, 84248640, 26127360, 3265920;

%F Formula

The row polynomials satisfy P(n,x) = -(3+4*x+2*x^2)*P(n-2,x) - (2*x+3*x^2+x^3)*d/dx P(n-2,x) + (4+4*x)*P(n-1,x) + (2*x+x^2)*d/dx 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[-(3 + 4*x + 2*x^2)*P[n - 2, x] - (2*x + 3*x^2 + x^3)*D[P[n - 2, x], x] + (4 + 4*x)*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], x]];
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] = -(3 + 4*x + 2*x^2)*v[i - 2] - (2*x + 3*x^2 + x^3)*deriv(v[i - 2], x) + (4 + 4*x)*v[i - 1] + (2*x + x^2)*deriv(v[i - 1], 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, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.