A367270 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n - 3\,n^{2} + 2\,n^{2} x + 2\,n^{3} - 4\,n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(-2 - 8\,x - 8\,x^{2} + 10\,n + 32\,n x + 20\,n x^{2} - 16\,n^{2} - 36\,n^{2} x - 8\,n^{2} x^{2} + 8\,n^{3} + 4\,n^{3} x - 16\,n^{3} x^{2} + 8\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x - 4\,x^{2} - 4\,x^{3} + 4\,n x + 12\,n x^{2} + 8\,n x^{3} - 4\,n^{2} x - 8\,n^{2} x^{2} - 4\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	3	8						11
3	10	48	36					94
4	35	240	360	128				763
5	126	1120	2520	1920	400			6086
6	462	5040	15120	17920	8400	1152		48094
7	1716	22176	83160	134400	100800	32256	3136	377644

All 15 rows

Technical details

Node id: n-ecbc0f405cc9cd9c · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3f6c9c5e8cc50607

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:     3,      8;
  3:    10,     48,      36;
  4:    35,    240,     360,     128;
  5:   126,   1120,    2520,    1920,     400;
  6:   462,   5040,   15120,   17920,    8400,    1152;
  7:  1716,  22176,   83160,  134400,  100800,   32256,    3136;
  8:  6435,  96096,  432432,  887040,  924000,  483840,  112896,   8192;
  9: 24310, 411840, 2162160, 5381376, 7207200, 5322240, 2069760, 368640, 20736;

%F Formula

The row polynomials satisfy (2*n^3-3*n^2+n+(2*n^4-4*n^3+2*n^2)*x)*P(n,x) = (8*n^3-16*n^2+10*n-2+(8*n^4+4*n^3-36*n^2+32*n-8)*x+(12*n^4-16*n^3-8*n^2+20*n-8)*x^2)*P(n-1,x) - ((4*n^2-4*n+1)*x+(4*n^3+8*n^2-12*n+4)*x^2+(8*n^3-8*n+4)*x^3)*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[((8*n^3 - 16*n^2 + 10*n - 2 + (8*n^4 + 4*n^3 - 36*n^2 + 32*n - 8)*x + (12*n^4 - 16*n^3 - 8*n^2 + 20*n - 8)*x^2)*P[n - 1, x] - ((4*n^2 - 4*n + 1)*x + (4*n^3 + 8*n^2 - 12*n + 4)*x^2 + (8*n^3 - 8*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + n + (2*n^4 - 4*n^3 + 2*n^2)*x)]];
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] = ((8*n^3 - 16*n^2 + 10*n - 2 + (8*n^4 + 4*n^3 - 36*n^2 + 32*n - 8)*x + (12*n^4 - 16*n^3 - 8*n^2 + 20*n - 8)*x^2)*v[i - 1] - ((4*n^2 - 4*n + 1)*x + (4*n^3 + 8*n^2 - 12*n + 4)*x^2 + (8*n^3 - 8*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + n + (2*n^4 - 4*n^3 + 2*n^2)*x)); 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 ≥ 2
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 basis_map_A013609 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