A290319 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	8	12													21
3	1	18	92	120												231
4	1	32	344	1408	1680											3465
5	1	50	920	7600	27024	30240										65835
6	1	72	2020	27840	194224	624768	665280									1514205
7	1	98	3892	80360	918064	5674592	16909248	17297280								40883535
8	1	128	6832	197120	3328864	33216512	187147008	524574720	518918400							1267389585
9	1	162	11184	429408	10030944	146397888	1316508416	6887572992	18354458880	17643225600						44358635475
10	1	200	17340	854400	26348448	527573760	6879628160	56914892800	280082232576	715112663040	670442572800					1729986783525
11	1	242	25740	1582680	62233248	1634208576	29037726080	345859275520	2670507730176	12478566431232	30705174420480	28158588057600				74389431691575
12	1	288	36872	2766720	135036528	4496937984	104211320576	1681594675200	18580034404096	135321922019328	604719230257152	1440596611399680	1295295050649600			3496303289504025
13	1	338	51272	4610320	273372528	11248764384	329058219776	6892160704000	102659768164096	1064323642224128	7370815331223552	31676558124257280	73325125620633600	64764752532480000		178311467764705275
14	1	392	69524	7379008	522329808	26010880896	936491496512	24661304571904	474836446180096	6607951123085312	64844292011326464	429700586010329088	1783859264330526720	4024321536046694400	3497296636753920000	9807130727058790125

Technical details

Node id: n-b1607adc23dda083 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-68e273231fc13fce

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,   8,    12;
  3: 1,  18,    92,    120;
  4: 1,  32,   344,   1408,     1680;
  5: 1,  50,   920,   7600,    27024,     30240;
  6: 1,  72,  2020,  27840,   194224,    624768,     665280;
  7: 1,  98,  3892,  80360,   918064,   5674592,   16909248,   17297280;
  8: 1, 128,  6832, 197120,  3328864,  33216512,  187147008,  524574720,   518918400;
  9: 1, 162, 11184, 429408, 10030944, 146397888, 1316508416, 6887572992, 18354458880, 17643225600;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-2)*x)*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[(1 + (4*n - 2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (4*n - 2)*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 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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A007318 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A290319 → 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
[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.