A386789 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	3	15	6						24
3	4	44	80	20					148
4	5	95	360	385	70				915
5	6	174	1092	2352	1764	252			5640
6	7	287	2660	9240	13650	7854	924		34622
7	8	440	5616	28200	66000	73656	34320	3432	211672

All 15 rows

Technical details

Node id: n-59227c6d4e642670 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8654f175b69333bf

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:  2,   2;
  2:  3,  15,     6;
  3:  4,  44,    80,     20;
  4:  5,  95,   360,    385,     70;
  5:  6, 174,  1092,   2352,   1764,     252;
  6:  7, 287,  2660,   9240,  13650,    7854,     924;
  7:  8, 440,  5616,  28200,  66000,   73656,   34320,    3432;
  8:  9, 639, 10710,  72765, 242550,  423423,  378378,  148005,  12870;
  9: 10, 890, 18920, 166320, 740740, 1821820, 2522520, 1876160, 632060, 48620;

%F Formula

The row polynomials satisfy (n^3-n^2+n*x-n*x^2)*P(n,x) = (n^3-n+(2*n^3+2*n^2-n-1)*x-(n-1)*x^2-2*n*x^3)*P(n-1,x) + ((2*n^2-2)*x+(2*n^2-4)*x^2-4*x^3-2*x^4)*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^3 - n + (2*n^3 + 2*n^2 - n - 1)*x - (n - 1)*x^2 - 2*n*x^3)*P[n - 1, x] + ((2*n^2 - 2)*x + (2*n^2 - 4)*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - n^2 + n*x - n*x^2)]];
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^3 - n + (2*n^3 + 2*n^2 - n - 1)*x - (n - 1)*x^2 - 2*n*x^3)*v[i - 1] + ((2*n^2 - 2)*x + (2*n^2 - 4)*x^2 - 4*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - n^2 + n*x - n*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

A357367 → D+1

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.