A069466 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n^{2} - 4\,n^{2} x + n^{2} x^{2} - 2\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(-12\,x^{2} + 4\,x^{3} - 2\,n + 4\,n x + 48\,n x^{2} - 14\,n x^{3} + 4\,n^{2} - 60\,n^{2} x^{2} + 12\,n^{2} x^{3} - 2\,n^{3} - 22\,n^{3} x + 24\,n^{3} x^{2} + 4\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x) + \left(8\,x^{2} - 12\,x^{3} + 4\,x^{4} + 2\,n x - 26\,n x^{2} + 32\,n x^{3} - 8\,n x^{4} - 8\,n^{2} x + 24\,n^{2} x^{2} - 16\,n^{2} x^{3} + 8\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	30	36	6						72
3	200	540	240	20					1000
4	1190	6160	5880	1400	70				14700
5	6552	56700	100800	50400	7560	252			222264
6	34188	449064	1316700	1201200	374220	38808	924		3415104
7	171600	3195192	14126112	21021000	11771760	2522520	192192	3432	53003808

All 15 rows

Technical details

Node id: n-c3c3ac76ac4313d2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a07eaa75d981d953

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:       4,         2;
  2:      30,        36,          6;
  3:     200,       540,        240,         20;
  4:    1190,      6160,       5880,       1400,         70;
  5:    6552,     56700,     100800,      50400,       7560,        252;
  6:   34188,    449064,    1316700,    1201200,     374220,      38808,       924;
  7:  171600,   3195192,   14126112,   21021000,   11771760,    2522520,    192192,     3432;
  8:  836550,  21003840,  131171040,  292612320,  264864600,  100900800,  15855840,   926640,   12870;
  9: 3986840, 129961260, 1092199680, 3430627200, 4631346720, 2830267440, 784143360, 94517280, 4375800, 48620;

%F Formula

The row polynomials satisfy (n^4-2*n^3+2*n^2+(2*n^3-4*n^2)*x+n^2*x^2)*P(n,x) = (4*n^4-2*n^3+4*n^2-2*n+(12*n^4-22*n^3+4*n)*x+(24*n^3-60*n^2+48*n-12)*x^2+(12*n^2-14*n+4)*x^3)*P(n-1,x) + ((8*n^3-8*n^2+2*n)*x-(8*n^3-24*n^2+26*n-8)*x^2-(16*n^2-32*n+12)*x^3-(8*n-4)*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[((4*n^4 - 2*n^3 + 4*n^2 - 2*n + (12*n^4 - 22*n^3 + 4*n)*x + (24*n^3 - 60*n^2 + 48*n - 12)*x^2 + (12*n^2 - 14*n + 4)*x^3)*P[n - 1, x] + ((8*n^3 - 8*n^2 + 2*n)*x - (8*n^3 - 24*n^2 + 26*n - 8)*x^2 - (16*n^2 - 32*n + 12)*x^3 - (8*n - 4)*x^4)*D[P[n - 1, x], x])/(n^4 - 2*n^3 + 2*n^2 + (2*n^3 - 4*n^2)*x + n^2*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] = ((4*n^4 - 2*n^3 + 4*n^2 - 2*n + (12*n^4 - 22*n^3 + 4*n)*x + (24*n^3 - 60*n^2 + 48*n - 12)*x^2 + (12*n^2 - 14*n + 4)*x^3)*v[i - 1] + ((8*n^3 - 8*n^2 + 2*n)*x - (8*n^3 - 24*n^2 + 26*n - 8)*x^2 - (16*n^2 - 32*n + 12)*x^3 - (8*n - 4)*x^4)*deriv(v[i - 1], x))/(n^4 - 2*n^3 + 2*n^2 + (2*n^3 - 4*n^2)*x + n^2*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly 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
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k …?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A008459 → 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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023