A306364 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	12	6							18
3	60	90							150
4	360	1080	135						1575
5	2520	12600	4725						19845
6	20160	151200	113400	6300					291060
7	181440	1905120	2381400	396900					4864860
8	1814400	25401600	47628000	15876000	496125				91216125
9	19958400	359251200	943034400	523908000	49116375				1895268375
10	239500800	5388768000	18860688000	15717240000	2946982500	58939650			43212118950
11	3113510400	85621536000	385296912000	449513064000	140472832500	8428369950			1072446224850
12	43589145600	1438441804800	8091235152000	12586365792000	5899858965000	707983075800	9833098275		28777307033475
13	653837184000	25499650176000	175310094960000	350620189920000	230094499635000	46018899927000	1917454163625		830114625965625
14	10461394944000	475993469952000	3926946127104000	9817365317760000	8590194653040000	2577058395912000	214754866326000	2191376187000	25614965601225000

Technical details

Node id: n-4c74df055ec16506 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-49471ab03f05f98d

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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:        3;
  2:       12,         6;
  3:       60,        90;
  4:      360,      1080,       135;
  5:     2520,     12600,      4725;
  6:    20160,    151200,    113400,      6300;
  7:   181440,   1905120,   2381400,    396900;
  8:  1814400,  25401600,  47628000,  15876000,   496125;
  9: 19958400, 359251200, 943034400, 523908000, 49116375;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 2*n + (n^2 + n - 2)*x)*v[i - 1] + ((2*n + 4)*x - (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 …?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.