A185410 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(n - 8\,n x - 8\,n x^{2} + 4\,n^{2} x + 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,n x + 2\,n x^{2} - 4\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-1 + 4\,x - 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	1	4													5
3	1	20	12												33
4	1	72	180	32											285
5	1	232	1548	1184	80										3045
6	1	716	10524	20672	6640	192									38745
7	1	2172	63360	256960	214800	33984	448								571725
8	1	6544	355896	2643520	4502800	1920384	164416	1024							9594585
9	1	19664	1916904	24198976	73569200	64351104	15580096	766976	2304						180405225
10	1	59028	10056456	205175040	1025107440	1590692544	803151552	118339584	3490560	5120					3756077325
11	1	177124	51876036	1650763008	12816530160	32259165120	28999049856	9118193664	858131712	15610880	11264				85769508825
12	1	531416	264693900	12800365920	148364031600	570168523008	828615959808	468190617600	96651982080	6017484800	68924416	24576			2131143139125
13	1	1594296	1341006228	96661065440	1623284517600	9120443159808	20084760019968	18386735284224	6910228143360	973460940800	41162812416	301326336	53248		57238379923725
14	1	4782940	6762425796	715963640768	17026087294400	135387820338048	431350152576000	597157189582848	365694804774144	95276070579200	9430875339776	276359823360	1307078656	114688	1652323398350625

Technical details

Node id: n-8854f7202c6a83aa · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6c0433a6238a5048

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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: 1;
  1: 1;
  2: 1,     4;
  3: 1,    20,      12;
  4: 1,    72,     180,       32;
  5: 1,   232,    1548,     1184,       80;
  6: 1,   716,   10524,    20672,     6640,      192;
  7: 1,  2172,   63360,   256960,   214800,    33984,      448;
  8: 1,  6544,  355896,  2643520,  4502800,  1920384,   164416,   1024;
  9: 1, 19664, 1916904, 24198976, 73569200, 64351104, 15580096, 766976, 2304;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + (4*n^2 - 8*n)*x - (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] + (2*n*x - (4*n^2 - 2*n)*x^2 + (4*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) - (1 - (4*n^2 - 8*n + 4)*x)*v[i - 1] + ((2*n - 2)*x - (2*n - 2)*x^2)*deriv(v[i - 1], x))/(n - 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 A decomposition of the double factorials A001147.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A triangular decomposition of the double factorial numbers A001147.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)

References

[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.