A370258 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 10\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(20 + 60\,x + 18\,n + 66\,n x + 4\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(14\,x + 15\,x^{2} + 8\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 10 + 30\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	10	30								40
1	21	168	252							441
2	36	540	1980	1980						4536
3	55	1320	8580	20020	15015					44990
4	78	2730	27300	109200	185640	111384				436332
5	105	5040	71400	428400	1220940	1627920	813960			4167765
6	136	8568	162792	1356600	5697720	12534984	13728792	5883768		39373360
7	171	13680	335160	3686760	21198870	67836384	121136400	112483800	42181425	368872650

All 15 rows

Technical details

Node id: n-0e4c3979a5ed69fd · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-23ffd7df897ade57

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:  10,    30;
  1:  21,   168,     252;
  2:  36,   540,    1980,     1980;
  3:  55,  1320,    8580,    20020,     15015;
  4:  78,  2730,   27300,   109200,    185640,     111384;
  5: 105,  5040,   71400,   428400,   1220940,    1627920,     813960;
  6: 136,  8568,  162792,  1356600,   5697720,   12534984,   13728792,    5883768;
  7: 171, 13680,  335160,  3686760,  21198870,   67836384,  121136400,  112483800,   42181425;
  8: 210, 20790,  637560,  8925840,  66943800,  290089800,  745945200, 1118917800,  901350450,  300450150;
  9: 253, 30360, 1138500, 19734000, 186486300, 1044323280, 3605401800, 7725861000, 9979237125, 7096346400, 2128903920;

%F Formula

The row polynomials satisfy (4*n^2+10*n+6)*P(n,x) = (4*n^2+18*n+20+(18*n^2+66*n+60)*x)*P(n-1,x) + ((8*n+14)*x+(9*n+15)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 10 + 30*x.

%t Mathematica

P[0, x_] := 10 + 30*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 18*n + 20 + (18*n^2 + 66*n + 60)*x)*P[n - 1, x] + ((8*n + 14)*x + (9*n + 15)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 10*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 10 + 30*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 18*n + 20 + (18*n^2 + 66*n + 60)*x)*v[i - 1] + ((8*n + 14)*x + (9*n + 15)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 10*n + 6)); 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) = binomial(n, k)*binomial(2*n+k, k), 0 <= k <= n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - 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 ?

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.