A112486 → D

nonnegative coefficients

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

Recurrence

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

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

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

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	5	6													11
3	26	70	45												141
4	154	680	945	420											2199
5	1044	6608	14700	13860	4725										40937
6	8028	67480	211596	315000	225225	62370									889699
7	69264	735768	3026940	6286280	7033950	4054050	945945								22152197
8	663696	8604432	44332860	119830480	185735550	166486320	80405325	16216200							622274863
9	6999840	107922672	673967448	2267122000	4547543000	5617652040	4199995800	1745944200	310134825						19477281825
10	80627040	1449068544	10704184920	43421810432	107887780000	171738206640	176262766680	113026914000	41247931725	6547290750					672366580731
11	1007441280	20772940032	178115425488	851138608320	2540363425600	4990994888880	6578870898600	5773488280560	3240908921250	1054113810750	151242416325				25381017057085
12	13575738240	317014633728	3107963122992	17180526242880	60208950401600	141661321882080	229809439739880	258325247980800	197962573658850	98811711999000	28988129795625	3794809718700			1040181264914375
13	196287356160	5135340176640	56860239226752	358372159419456	1448847289890000	3992078444434080	7736348096517040	10677617401651200	10458110604041100	7111754509860000	3195651428669700	853832186707500	102776096548125		45997580084497753
14	3031488633600	88054063511040	1089872231094144	7739530958689920	35591472094847760	112875287098173600	255611807063653040	420579937152735360	504807574583812500	437944934784357000	267641604777046500	109353766727205000	26824561199060625	2988412653476250	2183139846876296339

Technical details

Node id: n-fd5f222f6785be4b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-49ab8a1f300b12f8

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:       5,         6;
  3:      26,        70,        45;
  4:     154,       680,       945,        420;
  5:    1044,      6608,     14700,      13860,       4725;
  6:    8028,     67480,    211596,     315000,     225225,      62370;
  7:   69264,    735768,   3026940,    6286280,    7033950,    4054050,     945945;
  8:  663696,   8604432,  44332860,  119830480,  185735550,  166486320,   80405325,   16216200;
  9: 6999840, 107922672, 673967448, 2267122000, 4547543000, 5617652040, 4199995800, 1745944200, 310134825;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+2*n^2*x+n^2*x^2)*P(n-2,x) - (n*x+2*n*x^2+n*x^3)*d/dx P(n-2,x) + (2*n+1+(2*n+2)*x)*P(n-1,x) + (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[-(n^2 + 2*n^2*x + n^2*x^2)*P[n - 2, x] - (n*x + 2*n*x^2 + n*x^3)*D[P[n - 2, x], x] + (2*n + 1 + (2*n + 2)*x)*P[n - 1, x] + (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] = -(n^2 + 2*n^2*x + n^2*x^2)*v[i - 2] - (n*x + 2*n*x^2 + n*x^3)*deriv(v[i - 2], x) + (2*n + 1 + (2*n + 2)*x)*v[i - 1] + (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 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.?
  - 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.