A272867 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(22 + 20\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	22	20	4													46
2	484	880	576	160	16											2116
3	10648	29040	32208	18560	5856	960	64									97336
4	234256	851840	1331968	1168640	628864	212480	44032	5120	256							4477456
5	5153632	23425600	47277120	55756800	42535680	21926400	7733760	1843200	284160	25600	1024					205962976
6	113379904	618435840	1529223168	2265894400	2240029440	1556121600	778813440	282931200	74050560	13619200	1671168	122880	4096			9474296896
7	2494357888	15873186560	46465146112	82907883520	100715428352	88098841600	57216445440	28025241600	10402990080	2912358400	605351936	90603520	9232384	573440	16384	435817657216

All 15 rows

Technical details

Node id: n-c6c043a068fbd531 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-9217fe7884f21aef

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:            22,            20,              4;
  2:           484,           880,            576,            160,              16;
  3:         10648,         29040,          32208,          18560,            5856,             960,              64;
  4:        234256,        851840,        1331968,        1168640,          628864,          212480,           44032,            5120,            256;
  5:       5153632,      23425600,       47277120,       55756800,        42535680,        21926400,         7733760,         1843200,         284160,          25600,           1024;
  6:     113379904,     618435840,     1529223168,     2265894400,      2240029440,      1556121600,       778813440,       282931200,       74050560,       13619200,        1671168,        122880,          4096;
  7:    2494357888,   15873186560,    46465146112,    82907883520,    100715428352,     88098841600,     57216445440,     28025241600,    10402990080,     2912358400,      605351936,      90603520,       9232384,       573440,       16384;
  8:   54875873536,  399097262080,  1349674377216,  2816769105920,   4059757678592,   4284114616320,   3423600345088,   2113279590400,  1018236395520,   384232652800,   113176870912,   25749749760,    4436590592,    559677440,    48758784,    2621440,     65536;
  9: 1207269217792, 9877657236480, 37894285034496, 90558796922880, 151048748556288, 186712751554560, 177240530632704, 132100616355840, 78361193889792, 37270964633600, 14247489798144, 4366962524160, 1065307471872, 204043714560, 30012604416, 3271557120, 248905728, 11796480, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (22+20*x+4*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[(22 + 20*x + 4*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] = (22 + 20*x + 4*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 (−∞, -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)
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 basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

A272867 → basis A013610 → x ↦ x/(x+1)

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.