A282629 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 3\,x\right)\,P_{n-1}(x) + \left(3 + 3\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	25	33	9													67
3	199	360	189	27												775
4	1876	4251	3213	918	81											10339
5	20257	54663	53145	22545	4050	243										154903
6	245017	762282	898344	501120	136080	16767	729									2560339
7	3266914	11461089	15780366	10842552	3932145	739935	66339	2187								46091527
8	47450923	184710561	289769895	235480014	106541001	27049302	3725190	253692	6561							894987139
9	743935375	3173946066	5571150759	5209041879	2816835588	900613161	168429618	17675334	944784	19683						18602592247
10	12497579698	57876333141	112114722699	118233023760	74205692460	28593889947	6778473093	969847020	80011395	3444525	59049					411353076787
11	223619318215	1115315407275	2358873440253	2763841786497	1970898499845	887913502047	255274865307	46501868259	5242862295	348585930	12341241	177147				9627842654311
12	4240423494685	22631306447088	51809256702828	66657345543360	53144603887716	27377999613954	9256326776334	2054199997476	295717569147	26904987945	1471205835	43578162	531441			237495600335971
13	84855613320004	481995955830639	1185902596259784	1659708508824756	1460955693042846	846229686302286	328911387868206	86221201925430	15169126603059	1765343508786	132174041571	6045141375	151992126	1594323		6151853486255191
14	1785410320771933	10743954108333189	28242390409512561	42665386720015332	41047861909694094	26381636100499680	11585384832441420	3506323250213784	731063425399164	104198249828673	9989437605147	625664072124	24277287762	524532267	4782969	166804249234990099

Technical details

Node id: n-edca08000e1b5ff8 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-19bf2e621be60215

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:         4,          3;
  2:        25,         33,          9;
  3:       199,        360,        189,         27;
  4:      1876,       4251,       3213,        918,         81;
  5:     20257,      54663,      53145,      22545,       4050,       243;
  6:    245017,     762282,     898344,     501120,     136080,     16767,       729;
  7:   3266914,   11461089,   15780366,   10842552,    3932145,    739935,     66339,     2187;
  8:  47450923,  184710561,  289769895,  235480014,  106541001,  27049302,   3725190,   253692,   6561;
  9: 743935375, 3173946066, 5571150759, 5209041879, 2816835588, 900613161, 168429618, 17675334, 944784, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (4+3*x)*P(n-1,x) + (3+3*x)*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[(4 + 3*x)*P[n - 1, x] + (3 + 3*x)*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] = (4 + 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 - 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 (−∞, -1]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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

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.