A272866 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(19 + 36\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	19	36	16													71
2	361	1368	1904	1152	256											5041
3	6859	38988	91200	112320	76800	27648	4096									357911
4	130321	987696	3246112	6041088	6961920	5087232	2301952	589824	65536							25411681
5	2476099	23457780	99318320	247443840	401693440	443943936	338268160	175472640	59310080	11796480	1048576					1804229351
6	47045881	534837384	2771145744	8652216960	18129246720	26855000064	28836171776	22614736896	12856197120	5166858240	1393557504	226492416	16777216			128100283921
7	893871739	11855562012	72658649056	272710767168	700273830144	1301333354496	1804735213568	1897462185984	1519777021952	922829193216	418183643136	137141157888	30769414144	4227858432	268435456	9095120158391

All 15 rows

Technical details

Node id: n-414aa3fe097fc93d · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-67cad0a0ca30ceab

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:           19,            36,             16;
  2:          361,          1368,           1904,            1152,             256;
  3:         6859,         38988,          91200,          112320,           76800,            27648,             4096;
  4:       130321,        987696,        3246112,         6041088,         6961920,          5087232,          2301952,           589824,            65536;
  5:      2476099,      23457780,       99318320,       247443840,       401693440,        443943936,        338268160,        175472640,         59310080,         11796480,          1048576;
  6:     47045881,     534837384,     2771145744,      8652216960,     18129246720,      26855000064,      28836171776,      22614736896,      12856197120,       5166858240,       1393557504,        226492416,         16777216;
  7:    893871739,   11855562012,    72658649056,    272710767168,    700273830144,    1301333354496,    1804735213568,    1897462185984,    1519777021952,     922829193216,     418183643136,     137141157888,      30769414144,       4227858432,       268435456;
  8:  16983563041,  257435060832,  1821616512320,   7986904934400,  24285328775680,   54298563895296,   92342351101952,  121843582894080,  126060165529600,  102605122437120,   65483772526592,   32425560244224,   12212638842880,    3382286745600,    649613803520,    77309411328,     4294967296;
  9: 322687697779, 5502674425284, 44150112932688, 221448349170432, 778095688573440, 2033735028885504, 4097818231578624, 6508129736982528, 8258989746880512, 8437160611676160, 6954938734215168, 4615183414591488, 2447100667232256, 1022727410417664, 329507206594560, 78971563671552, 13258564042752, 1391569403904, 68719476736;

%F Formula

The row polynomials satisfy P(n,x) = (19+36*x+16*x^2)*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[(19 + 36*x + 16*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (19 + 36*x + 16*x^2)*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 basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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.