A269944 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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	1														2
2	1	3	2													6
3	1	8	17	10												36
4	1	18	97	180	100											396
5	1	35	403	1829	3160	1700										7128
6	1	61	1313	12307	50714	83860	44200									192456
7	1	98	3570	60888	506073	1960278	3147020	1635400								7313328
8	1	148	8470	239388	3550473	27263928	101160920	158986400	81770000							372979728
9	1	213	18090	789938	19110693	258044673	1873316240	6734446200	10415886000	5315050000						24616662048
10	1	295	35556	2273318	83885609	1825121499	23032979426	160346377880	562640474400	859417702000	435834100000					2043182949984
11	1	396	65351	5864474	313490727	10297568008	207370250825	2486677299906	16757624640280	57686105616400	87237022002000	44019244100000				208404660898368
12	1	518	113663	13837296	1028956555	48543436702	1463673547801	27785847900556	320132255228812	2102116311730560	7124941907202800	10686935928344000	5370347780200000			25633773290499264
13	1	663	188773	30318431	3035364475	197742137177	8502471869591	240018512331701	4349080200809432	48521293319908300	311931807108134000	1043803512472750000	1554976057390080000	778700428129000000		3742530900412892544
14	1	833	301483	62409841	8189497745	713754097927	42118635189681	1685438730162171	45152227297198602	787864927457511740	8560551671492545000	54072210720855530000	179001573177757580000	265124630184442600000	132379072781930000000	639972783970604625024

Technical details

Node id: n-39aebdc41ec576d0 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-e90640f2cfea9e20

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,   1;
  2: 1,   3,     2;
  3: 1,   8,    17,     10;
  4: 1,  18,    97,    180,      100;
  5: 1,  35,   403,   1829,     3160,      1700;
  6: 1,  61,  1313,  12307,    50714,     83860,      44200;
  7: 1,  98,  3570,  60888,   506073,   1960278,    3147020,    1635400;
  8: 1, 148,  8470, 239388,  3550473,  27263928,  101160920,  158986400,    81770000;
  9: 1, 213, 18090, 789938, 19110693, 258044673, 1873316240, 6734446200, 10415886000, 5315050000;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n^2-2*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 + (n^2 - 2*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 + (n^2 - 2*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, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k >…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
    
    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
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A269944 → 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.