A282629 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	1														4
2	9	15	1													25
3	27	108	63	1												199
4	81	594	945	255	1											1876
5	243	2835	8775	7380	1023	1										20257
6	729	12393	63180	109890	54729	4095	1									245017
7	2187	51030	387828	1151010	1263087	395388	16383	1								3266914
8	6561	201204	2133054	9658278	18752391	13817034	2816865	65535	1							47450923
9	19683	767637	10825650	69502860	210789621	285232185	146620935	19914660	262143	1						743935375
10	59049	2854035	51667875	446672880	1952923203	4228330491	4147881210	1525953330	140189049	1048575	1					12497579698
11	177147	10392624	234916605	2631715515	15685572969	49790532330	80096931486	58500315720	15680100447	984469068	4194303	1				223619318215
12	531441	37200870	1026921159	14472811485	112849606782	494454202308	1186310908728	1457051850936	807544405701	159754411674	6903866385	16777215	1			4240423494685
13	1594323	131265927	4345593057	75252990384	743787541926	4304602776474	14436925176960	26911062818640	25735462832079	10977340509135	1618255715895	48377396340	67108863	1		84855613320004
14	4782969	457570701	17893618470	373509135090	4564205327682	33739859503350	150925844942730	398345542349040	588516582050397	444699426840669	147560193766440	16327689347970	338843100969	268435455	1	1785410320771933

Technical details

Node id: n-9797801130dd5d88 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-537eddf8e19d10b5

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:     1;
  1:     3,      1;
  2:     9,     15,        1;
  3:    27,    108,       63,        1;
  4:    81,    594,      945,      255,         1;
  5:   243,   2835,     8775,     7380,      1023,         1;
  6:   729,  12393,    63180,   109890,     54729,      4095,         1;
  7:  2187,  51030,   387828,  1151010,   1263087,    395388,     16383,        1;
  8:  6561, 201204,  2133054,  9658278,  18752391,  13817034,   2816865,    65535,      1;
  9: 19683, 767637, 10825650, 69502860, 210789621, 285232185, 146620935, 19914660, 262143, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3+(3*n-2)*x)*P(n-1,x) - 3*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + (3*n - 2)*x)*P[n - 1, x] - 3*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (3*n - 2)*x)*v[i - 1] - 3*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A282629: Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.