A159854 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 4 + 4\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	4	4	1															9
1	12	16	7	1														36
2	36	60	37	10	1													144
3	108	216	171	67	13	1												576
4	324	756	729	372	106	16	1											2304
5	972	2592	2943	1845	690	154	19	1										9216
6	2916	8748	11421	8478	3915	1152	211	22	1									36864
7	8748	29160	43011	36855	20223	7371	1785	277	25	1								147456
8	26244	96228	158193	153576	97524	42336	12726	2616	352	28	1							589824
9	78732	314928	570807	618921	446148	224532	80514	20574	3672	436	31	1						2359296
10	236196	1023516	2027349	2427570	1957365	1119744	466074	142236	31590	4980	529	34	1					9437184
11	708588	3306744	7105563	9310059	8299665	5316597	2517966	892782	237006	46530	6567	631	37	1				37748736
12	2125764	10628820	24623433	35035740	34209054	24249456	12870495	5196312	1603800	376596	66231	8460	742	40	1			150994944
13	6377292	34012224	84499119	129730653	137662902	106957422	62860941	28459431	10007712	2733588	575289	91611	10686	862	43	1		603979776
14	19131876	108413964	287509581	473691078	542719359	458535168	295540245	148239234	58482567	18208476	4459455	850122	123669	13272	991	46	1	2415919104

Technical details

Node id: n-b5ecca697c4721f9 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-8987aa5000ed2f45

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     4,      4,      1;
  1:    12,     16,      7,      1;
  2:    36,     60,     37,     10,      1;
  3:   108,    216,    171,     67,     13,      1;
  4:   324,    756,    729,    372,    106,     16,     1;
  5:   972,   2592,   2943,   1845,    690,    154,    19,     1;
  6:  2916,   8748,  11421,   8478,   3915,   1152,   211,    22,    1;
  7:  8748,  29160,  43011,  36855,  20223,   7371,  1785,   277,   25,   1;
  8: 26244,  96228, 158193, 153576,  97524,  42336, 12726,  2616,  352,  28,  1;
  9: 78732, 314928, 570807, 618921, 446148, 224532, 80514, 20574, 3672, 436, 31, 1;

%F Formula

The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 4*x + x^2.

%t Mathematica

P[0, x_] := 4 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 4*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + x)*v[i - 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 (−∞, -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), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

A024462 → basis A013611

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.