A143491 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 36\,x + 81\,x^{2} - 6\,n - 12\,n x - n^{2}\right)\,P_{n-2}(x) + \left(36\,x^{2} + 36\,x^{3} + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(7 + 12\,x + 2\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 12 + 84\,x + 81\,x^{2}\)

\(P_{1}(x) = 60 + 564\,x + 972\,x^{2} + 432\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	12	84	81								177
1	60	564	972	432							2028
2	360	4104	9639	7776	2025						23904
3	2520	33048	95175	105840	50625	8748					295956
4	20160	294624	984474	1354320	901125	288684	35721				3879108
5	181440	2893536	10848978	17439408	14458500	6491016	1500282	139968			53953128
6	1814400	31112640	128021148	232255296	226332225	127370880	41507802	7278336	531441		796224168
7	19958400	364011840	1618242948	3237587712	3578351175	2378834892	976683582	242704512	33480783	1968300	12451824144

All 15 rows

Technical details

Node id: n-4e0745b0dd22f6a8 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-11954419b7b7488a

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,          84,           81;
  1:         60,         564,          972,          432;
  2:        360,        4104,         9639,         7776,         2025;
  3:       2520,       33048,        95175,       105840,        50625,         8748;
  4:      20160,      294624,       984474,      1354320,       901125,       288684,        35721;
  5:     181440,     2893536,     10848978,     17439408,     14458500,      6491016,      1500282,       139968;
  6:    1814400,    31112640,    128021148,    232255296,    226332225,    127370880,     41507802,      7278336,      531441;
  7:   19958400,   364011840,   1618242948,   3237587712,   3578351175,   2378834892,    976683582,    242704512,    33480783,    1968300;
  8:  239500800,  4607642880,  21875995296,  47481681600,  58116406500,  44004495780,  21433778793,   6739459200,  1323288090,  147622500,   7144929;
  9: 3113510400, 62773367040, 315489528288, 733933835712, 978083667000, 823121321220, 458324148744, 171598388544, 42791629320, 6820159500, 628753752, 25509168;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(12*n+36)*x-81*x^2)*P(n-2,x) + ((12*n+36)*x^2+36*x^3)*d/dx P(n-2,x) + (2*n+7+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 84*x + 81*x^2, P(1,x) = 60 + 564*x + 972*x^2 + 432*x^3.

%t Mathematica

P[0, x_] := 12 + 84*x + 81*x^2;
P[1, x_] := 60 + 564*x + 972*x^2 + 432*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (12*n + 36)*x - 81*x^2)*P[n - 2, x] + ((12*n + 36)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n + 7 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 84*x + 81*x^2; v[2] = 60 + 564*x + 972*x^2 + 432*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (12*n + 36)*x - 81*x^2)*v[i - 2] + ((12*n + 36)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n + 7 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 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 (−∞, 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_A013610 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)

Related ?

Same fingerprint ?

A136124 → basis A013610 → (θ+1)²

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.