A137320 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 8\,x + 7\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	6	42	72													120
3	24	228	708	720												1680
4	120	1404	6048	11388	7920											26880
5	720	9864	53136	140904	184176	95040										483840
6	5040	78408	500184	1677096	3120984	3059568	1235520									9676800
7	40320	697824	5099184	20419344	48447216	68170320	52718112	17297280								212889600
8	362880	6885216	56360016	260261856	742315104	1340241120	1497017808	946447200	259459200							5109350400
9	3628800	74658240	673763616	3504378816	11587340736	25279452864	36414036000	33416756928	17737747200	4151347200						132843110400
10	39916800	882930240	8680589856	50002148448	187035187968	475058774016	830305094688	986622938208	763200086976	347206521600	70572902400					3719607091200
11	479001600	11313665280	120059822592	756276398784	3144460927680	9067338671616	18514719068544	26784966962880	26917613931456	17904079824768	7096592217600	1270312243200				111588212736000
12	6227020800	156178679040	1775737334016	12112729813440	55247243636736	177620160356928	412970782651776	699984232819776	858843353403648	744187702419648	432433215499392	151349311296000	24135932620800			3570822807552000
13	87178291200	2311041922560	27983896257024	205092964068480	1015716007183104	3591627117731712	9333994164263424	18059194912512384	26023491604046592	27595494901948032	20937819065384448	10767554668131840	3364889282611200	482718652416000		121407975456768000
14	1307674368000	36496372953600	468290324229120	3664056282424704	19542692353184640	75204442916820864	215434081936317312	466901801137217664	769595467223458944	960425747214198912	893572678921675392	601207520395051008	276591987269936640	77903454721075200	10137091700736000	4370687116443648000

Technical details

Node id: n-3fff123d5c8ae651 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-2103b888984bf533

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:       2,        8;
  2:       6,       42,        72;
  3:      24,      228,       708,        720;
  4:     120,     1404,      6048,      11388,        7920;
  5:     720,     9864,     53136,     140904,      184176,       95040;
  6:    5040,    78408,    500184,    1677096,     3120984,     3059568,     1235520;
  7:   40320,   697824,   5099184,   20419344,    48447216,    68170320,    52718112,    17297280;
  8:  362880,  6885216,  56360016,  260261856,   742315104,  1340241120,  1497017808,   946447200,   259459200;
  9: 3628800, 74658240, 673763616, 3504378816, 11587340736, 25279452864, 36414036000, 33416756928, 17737747200, 4151347200;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+8)*x+(n+7)*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[Together[((n + 1 + (2*n + 8)*x + (n + 7)*x^2)*P[n - 1, x])/(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] = ((n + 1 + (2*n + 8)*x + (n + 7)*x^2)*v[i - 1])/(1 + x)); 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, 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_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    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 Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ 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.