Hadamard-factor parent of A155495 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9\,n - 2\,n x + n x^{2} - 21\,n^{2} x + 6\,n^{2} x^{2} - 22\,n^{3} x + 13\,n^{3} x^{2} + 12\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(9\,n + 106\,n x - 61\,n x^{2} + 22\,n x^{3} - 21\,n^{2} x - 245\,n^{2} x^{2} + 90\,n^{2} x^{3} - 22\,n^{3} x - 101\,n^{3} x^{2} + 127\,n^{3} x^{3} + 12\,n^{4} x^{2} + 68\,n^{4} x^{3} + 4\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(18\,x - 54\,x^{2} + 54\,x^{3} - 18\,x^{4} - 108\,n x^{2} + 178\,n x^{3} - 70\,n x^{4} - 44\,n^{2} x^{2} + 138\,n^{2} x^{3} - 94\,n^{2} x^{4} + 48\,n^{3} x^{3} - 48\,n^{3} x^{4} + 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1 + 24\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	1	24	9								34
2	1	60	135	16							212
3	1	112	630	448	25						1216
4	1	180	1890	3360	1125	36					6592
5	1	264	4455	14784	12375	2376	49				34304
6	1	364	9009	48048	75075	36036	4459	64			173056
7	1	480	16380	128128	321750	288288	89180	7680	81		851968
8	1	612	27540	297024	1093950	1575288	909636	195840	12393	100	4112384

All 15 rows

Technical details

Node id: n-293e5a07b1a24e54 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-395f2289776a8710

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,  24,     9;
   2: 1,  60,   135,      16;
   3: 1, 112,   630,     448,      25;
   4: 1, 180,  1890,    3360,    1125,       36;
   5: 1, 264,  4455,   14784,   12375,     2376,       49;
   6: 1, 364,  9009,   48048,   75075,    36036,     4459,       64;
   7: 1, 480, 16380,  128128,  321750,   288288,    89180,     7680,      81;
   8: 1, 612, 27540,  297024, 1093950,  1575288,   909636,   195840,   12393,    100;
   9: 1, 760, 43605,  620160, 3149250,  6651216,  6172530,  2480640,  392445,  19000,   121;
  10: 1, 924, 65835, 1193808, 7994250, 23279256, 31685654, 20465280, 6043653, 731500, 27951, 144;

%F Formula

The row polynomials satisfy (9*n-(22*n^3+21*n^2+2*n)*x+(4*n^5+12*n^4+13*n^3+6*n^2+n)*x^2)*P(n,x) = (9*n-(22*n^3+21*n^2-106*n)*x+(4*n^5+12*n^4-101*n^3-245*n^2-61*n)*x^2+(12*n^5+68*n^4+127*n^3+90*n^2+22*n)*x^3)*P(n-1,x) + (18*x-(44*n^2+108*n+54)*x^2+(8*n^4+48*n^3+138*n^2+178*n+54)*x^3-(8*n^4+48*n^3+94*n^2+70*n+18)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 24*x + 9*x^2.

%t Mathematica

P[1, x_] := 1 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - (22*n^3 + 21*n^2 - 106*n)*x + (4*n^5 + 12*n^4 - 101*n^3 - 245*n^2 - 61*n)*x^2 + (12*n^5 + 68*n^4 + 127*n^3 + 90*n^2 + 22*n)*x^3)*P[n - 1, x] + (18*x - (44*n^2 + 108*n + 54)*x^2 + (8*n^4 + 48*n^3 + 138*n^2 + 178*n + 54)*x^3 - (8*n^4 + 48*n^3 + 94*n^2 + 70*n + 18)*x^4)*D[P[n - 1, x], x])/(9*n - (22*n^3 + 21*n^2 + 2*n)*x + (4*n^5 + 12*n^4 + 13*n^3 + 6*n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 24*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = ((9*n - (22*n^3 + 21*n^2 - 106*n)*x + (4*n^5 + 12*n^4 - 101*n^3 - 245*n^2 - 61*n)*x^2 + (12*n^5 + 68*n^4 + 127*n^3 + 90*n^2 + 22*n)*x^3)*v[i - 1] + (18*x - (44*n^2 + 108*n + 54)*x^2 + (8*n^4 + 48*n^3 + 138*n^2 + 178*n + 54)*x^3 - (8*n^4 + 48*n^3 + 94*n^2 + 70*n + 18)*x^4)*deriv(v[i - 1], x))/(9*n - (22*n^3 + 21*n^2 + 2*n)*x + (4*n^5 + 12*n^4 + 13*n^3 + 6*n^2 + n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? 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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

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