A176230 → D

nonnegative coefficients

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

Recurrence

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

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

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

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	6	2													8
3	45	30	3												78
4	420	420	84	4											928
5	4725	6300	1890	180	5										13100
6	62370	103950	41580	5940	330	6									214176
7	945945	1891890	945945	180180	15015	546	7								3979528
8	16216200	37837800	22702680	5405400	600600	32760	840	8							82796288
9	310134825	827026200	578918340	165405240	22972950	1670760	64260	1224	9						1906193808
10	6547290750	19641872250	15713497800	5237832600	872972100	79361100	4069800	116280	1710	10					48097014400
11	151242416325	504141387750	453727248975	172848475800	33609425850	3666482820	235030950	8953560	197505	2310	11				1319479621856
12	3794809718700	13914302301900	13914302301900	5963272415100	1325171647800	168658209720	12973708440	617795640	18170460	318780	3036	12			39094126591488
13	102776096548125	411104386192500	452214824811750	215340392767500	53835098191875	7830559737000	702742540500	40156716600	1476349875	34534500	493350	3900	13		1243845768887488
14	2988412653476250	12949788165063750	15539745798076500	8139866846611500	2261074124058750	369993947573250	37948097187000	2529873145800	111612050550	3263510250	62162100	737100	4914	14	42289474443657728

Technical details

Node id: n-e818b4e18e54f95b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6c767e931fb09e25

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:         6,         2;
  3:        45,        30,         3;
  4:       420,       420,        84,         4;
  5:      4725,      6300,      1890,       180,        5;
  6:     62370,    103950,     41580,      5940,      330,       6;
  7:    945945,   1891890,    945945,    180180,    15015,     546,     7;
  8:  16216200,  37837800,  22702680,   5405400,   600600,   32760,   840,    8;
  9: 310134825, 827026200, 578918340, 165405240, 22972950, 1670760, 64260, 1224, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-n+n*x)*P(n-1,x) + 2*n*x*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[Together[((2*n^2 - n + n*x)*P[n - 1, x] + 2*n*x*D[P[n - 1, x], x])/(n - 1)]];
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] = ((2*n^2 - n + n*x)*v[i - 1] + 2*n*x*deriv(v[i - 1], x))/(n - 1)); 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 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].?
  - 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

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.