A367270 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + 11\,n x - 3\,n x^{2} + 10\,n^{2} - 17\,n^{2} x + 3\,n^{2} x^{2} - 10\,n^{3} + 6\,n^{3} x + 3\,n^{4}\right)\,P_{n}(x) = \left(-6 - 34\,x - 42\,x^{2} + 18\,x^{3} + 4\,n + 66\,n x + 138\,n x^{2} - 45\,n x^{3} + 24\,n^{2} + 7\,n^{2} x - 150\,n^{2} x^{2} + 27\,n^{2} x^{3} - 34\,n^{3} - 66\,n^{3} x + 54\,n^{3} x^{2} + 12\,n^{4} + 27\,n^{4} x\right)\,P_{n-1}(x) + \left(-2\,x - 11\,x^{2} - 12\,x^{3} + 9\,x^{4} + 16\,n x^{2} + 42\,n x^{3} - 18\,n x^{4} + 8\,n^{2} x + 12\,n^{2} x^{2} - 36\,n^{2} x^{3} - 6\,n^{3} x - 18\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	9	6						15
3	46	90	27					163
4	215	720	594	108				1637
5	966	4620	6750	3240	405			15981
6	4242	26460	56700	49140	15795	1458		153795
7	18348	141372	402570	521640	306180	71442	5103	1466655

All 15 rows

Technical details

Node id: n-5e66375cc3d4a1ac · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-3d42264d70e1ef8b

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:      9,       6;
  3:     46,      90,       27;
  4:    215,     720,      594,      108;
  5:    966,    4620,     6750,     3240,      405;
  6:   4242,   26460,    56700,    49140,    15795,     1458;
  7:  18348,  141372,   402570,   521640,   306180,    71442,    5103;
  8:  78507,  720720,  2569644,  4490640,  3997350,  1714608,  306180,   17496;
  9: 333190, 3552120, 15243228, 33729696, 40977090, 26943840, 8879220, 1259712, 59049;

%F Formula

The row polynomials satisfy (3*n^4-10*n^3+10*n^2-3*n+(6*n^3-17*n^2+11*n)*x+(3*n^2-3*n)*x^2)*P(n,x) = (12*n^4-34*n^3+24*n^2+4*n-6+(27*n^4-66*n^3+7*n^2+66*n-34)*x+(54*n^3-150*n^2+138*n-42)*x^2+(27*n^2-45*n+18)*x^3)*P(n-1,x) - ((6*n^3-8*n^2+2)*x+(18*n^3-12*n^2-16*n+11)*x^2+(36*n^2-42*n+12)*x^3+(18*n-9)*x^4)*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[((12*n^4 - 34*n^3 + 24*n^2 + 4*n - 6 + (27*n^4 - 66*n^3 + 7*n^2 + 66*n - 34)*x + (54*n^3 - 150*n^2 + 138*n - 42)*x^2 + (27*n^2 - 45*n + 18)*x^3)*P[n - 1, x] - ((6*n^3 - 8*n^2 + 2)*x + (18*n^3 - 12*n^2 - 16*n + 11)*x^2 + (36*n^2 - 42*n + 12)*x^3 + (18*n - 9)*x^4)*D[P[n - 1, x], x])/(3*n^4 - 10*n^3 + 10*n^2 - 3*n + (6*n^3 - 17*n^2 + 11*n)*x + (3*n^2 - 3*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^4 - 34*n^3 + 24*n^2 + 4*n - 6 + (27*n^4 - 66*n^3 + 7*n^2 + 66*n - 34)*x + (54*n^3 - 150*n^2 + 138*n - 42)*x^2 + (27*n^2 - 45*n + 18)*x^3)*v[i - 1] - ((6*n^3 - 8*n^2 + 2)*x + (18*n^3 - 12*n^2 - 16*n + 11)*x^2 + (36*n^2 - 42*n + 12)*x^3 + (18*n - 9)*x^4)*deriv(v[i - 1], x))/(3*n^4 - 10*n^3 + 10*n^2 - 3*n + (6*n^3 - 17*n^2 + 11*n)*x + (3*n^2 - 3*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013610 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
    
    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
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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
[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