A143491 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 6\,x - x^{2} - 6\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(7 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 12 + 14\,x + 3\,x^{2}\)

\(P_{1}(x) = 60 + 94\,x + 36\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	12	14	3								29
1	60	94	36	4							194
2	360	684	357	72	5						1478
3	2520	5508	3525	980	125	6					12664
4	20160	49104	36462	12540	2225	198	7				120696
5	181440	482256	401814	161476	35700	4452	294	8			1267440
6	1814400	5185440	4741524	2150512	558845	87360	8134	416	9		14546640
7	19958400	60668640	59934924	29977664	8835435	1631574	191394	13872	567	10	181212480

All 15 rows

Technical details

Node id: n-eea91c0fea9c11af · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c70223fe9dad004e

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:         12,          14,           3;
  1:         60,          94,          36,          4;
  2:        360,         684,         357,         72,          5;
  3:       2520,        5508,        3525,        980,        125,         6;
  4:      20160,       49104,       36462,      12540,       2225,       198,        7;
  5:     181440,      482256,      401814,     161476,      35700,      4452,      294,       8;
  6:    1814400,     5185440,     4741524,    2150512,     558845,     87360,     8134,     416,      9;
  7:   19958400,    60668640,    59934924,   29977664,    8835435,   1631574,   191394,   13872,    567,    10;
  8:  239500800,   767940480,   810222048,  439645200,  143497300,  30181410,  4200231,  385200,  22410,   750,  11;
  9: 3113510400, 10462227840, 11684797344, 6795683664, 2415021400, 564555090, 89814648, 9807864, 724680, 34650, 968, 12;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(2*n+6)*x+x^2)*P(n-2,x) + (2*n+7+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 12 + 14*x + 3*x^2, P(1,x) = 60 + 94*x + 36*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 12 + 14*x + 3*x^2;
P[1, x_] := 60 + 94*x + 36*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (2*n + 6)*x + x^2)*P[n - 2, x] + (2*n + 7 + 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] = 12 + 14*x + 3*x^2; v[2] = 60 + 94*x + 36*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (2*n + 6)*x + x^2)*v[i - 2] + (2*n + 7 + 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    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 Unsigned 2-Stirling numbers of the first kind.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A136124 → x ↦ x+1 → θ+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.