A159854 → x ↦ x+1 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

\(P_{0}(x) = 4 + 16\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	4	16	16															36
1	12	64	112	64														252
2	36	240	592	640	256													1764
3	108	864	2736	4288	3328	1024												12348
4	324	3024	11664	23808	27136	16384	4096											86436
5	972	10368	47088	118080	176640	157696	77824	16384										605052
6	2916	34992	182736	542592	1002240	1179648	864256	360448	65536									4235364
7	8748	116640	688176	2358720	5177088	7547904	7311360	4538368	1638400	262144								29647548
8	26244	384912	2531088	9828864	24966144	43352064	52125696	42860544	23068672	7340032	1048576							207532836
9	78732	1259712	9132912	39610944	114213888	229920768	329785344	337084416	240648192	114294784	32505856	4194304						1452729852
10	236196	4094064	32437584	155364480	501085440	1146617856	1909039104	2330394624	2070282240	1305477120	554696704	142606336	16777216					10169108964
11	708588	13226976	113689008	595843776	2124714240	5444195328	10313588736	14627340288	15532425216	12197560320	6885998592	2646605824	620756992	67108864				71183762748
12	2125764	42515280	393974928	2242287360	8757517824	24831442944	52717547520	85136375808	105106636800	98722381824	69448237056	35483811840	12448694272	2684354560	268435456			498286339236
13	6377292	136048896	1351985904	8302761792	35241702912	109524400128	257478414336	466279317504	655865413632	716593692672	603234238464	384244383744	179281330176	57847840768	11542724608	1073741824		3488004374652
14	19131876	433655856	4600153296	30316228992	138936155904	469540012032	1210532843520	2428751609856	3832713510912	4773242732544	4676077486080	3565670105088	2074821525504	890668843008	266019536896	49392123904	4294967296	24416030622564

Technical details

Node id: n-e2695402b3f168fd · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-201a8fcec63b5aa2

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,      16,      16;
  1:    12,      64,     112,       64;
  2:    36,     240,     592,      640,       256;
  3:   108,     864,    2736,     4288,      3328,      1024;
  4:   324,    3024,   11664,    23808,     27136,     16384,      4096;
  5:   972,   10368,   47088,   118080,    176640,    157696,     77824,     16384;
  6:  2916,   34992,  182736,   542592,   1002240,   1179648,    864256,    360448,     65536;
  7:  8748,  116640,  688176,  2358720,   5177088,   7547904,   7311360,   4538368,   1638400,    262144;
  8: 26244,  384912, 2531088,  9828864,  24966144,  43352064,  52125696,  42860544,  23068672,   7340032,  1048576;
  9: 78732, 1259712, 9132912, 39610944, 114213888, 229920768, 329785344, 337084416, 240648192, 114294784, 32505856, 4194304;

%F Formula

The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 16*x + 16*x^2.

%t Mathematica

P[0, x_] := 4 + 16*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 4*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 + 16*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 4*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 (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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)

Related ?

Same fingerprint ?

A024462 → basis A007318

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.