A075498 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

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	1														5
2	19	11	1													31
3	109	102	21	1												233
4	742	949	321	34	1											2047
5	5815	9311	4465	775	50	1										20417
6	51193	97782	60936	15140	1590	69	1									226711
7	498118	1101283	843402	276836	41615	2919	91	1								2764265
8	5296321	13267511	12026827	4941650	986461	98714	4942	116	1							36622543
9	60987817	170329860	178010631	88105809	22205736	2950983	209874	7866	144	1						522808721
10	754940848	2321360623	2743388451	1589854980	487662330	82052145	7733397	409980	11925	175	1					7987414855
11	9983845261	33464795915	44063939953	29263451151	10623234435	2185854231	260796459	18269097	748605	17380	209	1				129864952697
12	140329768789	508621056384	737475255804	551967618136	232023015576	56848801086	8307027366	735490404	39699297	1293655	24519	246	1			2236349051263
13	2087182244308	8125528698301	12851082177648	10697330478484	5117067883642	1461676728798	255048701622	27647070594	1881999405	80538172	2135419	33657	286	1		40624528690337
14	32725315072135	136072376198303	232912324781137	213321193001644	114495567548726	37479802360000	7653336646956	991393452192	82517584578	4442745307	154253099	3391024	45136	329	1	775738427080567

Technical details

Node id: n-c35b0a8845432676 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-3c95c131f610ea73

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,         1;
  2:       19,        11,         1;
  3:      109,       102,        21,        1;
  4:      742,       949,       321,       34,        1;
  5:     5815,      9311,      4465,      775,       50,       1;
  6:    51193,     97782,     60936,    15140,     1590,      69,      1;
  7:   498118,   1101283,    843402,   276836,    41615,    2919,     91,    1;
  8:  5296321,  13267511,  12026827,  4941650,   986461,   98714,   4942,  116,   1;
  9: 60987817, 170329860, 178010631, 88105809, 22205736, 2950983, 209874, 7866, 144, 1;

%F Formula

The row polynomials satisfy P(n,x) = (4+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 + 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 + 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 Stirling2 triangle with scaled diagonals (powers of 3).?
  - 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.