A143491 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 4\,x^{2} + 2\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 23\,x^{2} + 9\,x^{3} + n + 11\,n x + 23\,n x^{2} + 6\,n x^{3} + 2\,n^{2} x + 8\,n^{2} x^{2} + n^{2} x^{3} + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x + 5\,x^{2} + 9\,x^{3} + 2\,n x^{2} + 6\,n x^{3} + n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 12 + 32\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	12	32	9								53
1	60	240	135	16							451
2	360	1920	1620	384	25						4309
3	2520	16800	18900	6720	875	36					45851
4	20160	161280	226800	107520	21000	1728	49				538537
5	181440	1693440	2857680	1693440	441000	54432	3087	64			6924583
6	1814400	19353600	38102400	27095040	8820000	1451520	123480	5120	81		96765641
7	19958400	239500800	538876800	447068160	174636000	35925120	4074840	253440	8019	100	1460301679

All 15 rows

Technical details

Node id: n-244e4c812771bfc9 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-8360a1098ab5c82c

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,          32,            9;
  1:         60,         240,          135,           16;
  2:        360,        1920,         1620,          384,          25;
  3:       2520,       16800,        18900,         6720,         875,          36;
  4:      20160,      161280,       226800,       107520,       21000,        1728,         49;
  5:     181440,     1693440,      2857680,      1693440,      441000,       54432,       3087,        64;
  6:    1814400,    19353600,     38102400,     27095040,     8820000,     1451520,     123480,      5120,       81;
  7:   19958400,   239500800,    538876800,    447068160,   174636000,    35925120,    4074840,    253440,     8019,    100;
  8:  239500800,  3193344000,   8083152000,   7664025600,  3492720000,   862202880,  122245200,  10137600,   481140,  12000,   121;
  9: 3113510400, 45664819200, 128432304000, 136994457600, 71351280000, 20549168640, 3496212720, 362419200, 22934340, 858000, 17303, 144;

%F Formula

The row polynomials satisfy (1+(2*n+3)*x+(n^2+4*n+4)*x^2)*P(n,x) = (n+4+(2*n^2+11*n+16)*x+(n^3+8*n^2+23*n+23)*x^2+(n^2+6*n+9)*x^3)*P(n-1,x) + (x+(2*n+5)*x^2+(n^2+6*n+9)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12 + 32*x + 9*x^2.

%t Mathematica

P[0, x_] := 12 + 32*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (2*n^2 + 11*n + 16)*x + (n^3 + 8*n^2 + 23*n + 23)*x^2 + (n^2 + 6*n + 9)*x^3)*P[n - 1, x] + (x + (2*n + 5)*x^2 + (n^2 + 6*n + 9)*x^3)*D[P[n - 1, x], x])/(1 + (2*n + 3)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 32*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (2*n^2 + 11*n + 16)*x + (n^3 + 8*n^2 + 23*n + 23)*x^2 + (n^2 + 6*n + 9)*x^3)*v[i - 1] + (x + (2*n + 5)*x^2 + (n^2 + 6*n + 9)*x^3)*deriv(v[i - 1], x))/(1 + (2*n + 3)*x + (n^2 + 4*n + 4)*x^2)); 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_A008277 of Unsigned 2-Stirling numbers of the first kind.?
  - Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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
  - Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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 A008277 → (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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.