A137320 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36\,x^{2} - 12\,n x - n^{2}\right)\,P_{n-2}(x) + \left(1 + 12\,x + 2\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 6\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	6							6
2	30	72						102
3	156	648	648					1452
4	924	5112	9072	5184				20292
5	6264	41760	100440	103680	38880			291024
6	48168	367488	1078920	1529280	1049760	279936		4353552
7	415584	3517920	11938752	20865600	19867680	9797760	1959552	68362848

All 15 rows

Technical details

Node id: n-e38c32d951b1da96 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5544581df894f340

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:        0;
  1:        6;
  2:       30,        72;
  3:      156,       648,        648;
  4:      924,      5112,       9072,       5184;
  5:     6264,     41760,     100440,     103680,      38880;
  6:    48168,    367488,    1078920,    1529280,    1049760,     279936;
  7:   415584,   3517920,   11938752,   20865600,   19867680,    9797760,    1959552;
  8:  3982176,  36648288,  139110048,  283300416,  335301120,  231227136,   86220288,  13436928;
  9: 41999040, 414268992, 1720935072, 3945884544, 5477764320, 4726439424, 2480792832, 725594112, 90699264;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+12*n*x+36*x^2)*P(n-2,x) + (2*n+1+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 12*n*x + 36*x^2)*P[n - 2, x] + (2*n + 1 + 12*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] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 12*n*x + 36*x^2)*v[i - 2] + (2*n + 1 + 12*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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.