A225466 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(5 + 3\,x\right)\,P_{n-1}(x) + \left(3 + 3\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	3														8
3	34	39	9													82
4	287	468	216	27												998
5	2839	5901	4023	1026	81											13870
6	31898	79863	71190	27405	4455	243										215054
7	399079	1161738	1269324	650700	161595	18225	729									3661390
8	5480609	18107085	23304078	14856912	4972590	862407	71442	2187								67657310
9	81724300	301122975	444378321	337580082	142040871	33451866	4276314	271188	6561							1344852478
10	1311990425	5320426626	8829751194	7763746563	3929213043	1172133585	204405768	20037294	1003833	19683						28552728014
11	22521232003	99477891447	182962262079	182332261980	107669865195	38909614653	8638516593	1158278940	89754480	3641355	59049					643763377774
12	411039834356	1961160400059	3952008915030	4393576836297	2961028714050	1256695187319	339738583338	58184909739	6176033325	387558270	12990780	177147				15340010139710
13	7938680371957	40634456193720	88897770792180	108898447021848	82368646457526	40132284388614	12805956083250	2680248203028	364123768932	31319688015	1623197961	45703926	531441			384753602402398
14	161596770440945	882278315418711	2079864870491772	2779695328172724	2328946596672696	1280258836643358	471219055344180	116843291982846	19445965484517	2143297262106	152279222238	6625474947	158900859	1594323		10122451393106222
15	3454818798460858	20022206057623155	50542606475220141	73102688365965804	67135060677604374	41073959519080488	17132523914485980	4918285929214944	972331901076393	131491785508083	11978305237899	714326145624	26453538657	546852789	4782969	278498693060798158

Technical details

Node id: n-8c6d2d54c414406c · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-109452e8310de7ba

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:          1;
   2:          5,          3;
   3:         34,         39,          9;
   4:        287,        468,        216,         27;
   5:       2839,       5901,       4023,       1026,         81;
   6:      31898,      79863,      71190,      27405,       4455,        243;
   7:     399079,    1161738,    1269324,     650700,     161595,      18225,       729;
   8:    5480609,   18107085,   23304078,   14856912,    4972590,     862407,     71442,     2187;
   9:   81724300,  301122975,  444378321,  337580082,  142040871,   33451866,   4276314,   271188,    6561;
  10: 1311990425, 5320426626, 8829751194, 7763746563, 3929213043, 1172133585, 204405768, 20037294, 1003833, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (5+3*x)*P(n-1,x) + (3+3*x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 3*x)*P[n - 1, x] + (3 + 3*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (5 + 3*x)*v[i - 1] + (3 + 3*x)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >=…?
  - 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

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.