A086645 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-18 + 78\,x - 18\,x^{2} + 33\,n - 199\,n x + 69\,n x^{2} - 20\,n^{2} + 188\,n^{2} x - 104\,n^{2} x^{2} + 4\,n^{3} - 78\,n^{3} x + 77\,n^{3} x^{2} + 12\,n^{4} x - 28\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-72 - 120\,x + 40\,x^{2} + 8\,x^{3} + 132\,n - 4\,n x - 496\,n x^{2} - 60\,n x^{3} - 80\,n^{2} + 272\,n^{2} x + 726\,n^{2} x^{2} + 16\,n^{3} - 216\,n^{3} x - 324\,n^{3} x^{2} + 126\,n^{3} x^{3} + 48\,n^{4} x + 8\,n^{4} x^{2} - 104\,n^{4} x^{3} + 16\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-18\,x - 34\,x^{2} + 6\,x^{3} + 4\,x^{4} + 24\,n x - 8\,n x^{2} - 126\,n x^{3} - 28\,n x^{4} - 8\,n^{2} x + 56\,n^{2} x^{2} + 146\,n^{2} x^{3} + 4\,n^{2} x^{4} - 24\,n^{3} x^{2} - 32\,n^{3} x^{3} + 32\,n^{3} x^{4} - 8\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	8							10
3	8	64	36						108
4	32	384	648	128					1192
5	128	2048	5760	4096	400				12432
6	512	10240	40320	51200	20000	1152			123424
7	2048	49152	248832	458752	336000	82944	3136		1180864
8	8192	229376	1419264	3440640	3763200	1806336	307328	8192	10982528

All 15 rows

Technical details

Node id: n-f6a165d0ec5e0cba · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-beb8457930cf3c20

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 -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;
   2:      2,       8;
   3:      8,      64,       36;
   4:     32,     384,      648,       128;
   5:    128,    2048,     5760,      4096,       400;
   6:    512,   10240,    40320,     51200,     20000,      1152;
   7:   2048,   49152,   248832,    458752,    336000,     82944,      3136;
   8:   8192,  229376,  1419264,   3440640,   3763200,   1806336,    307328,     8192;
   9:  32768, 1048576,  7667712,  23068672,  33792000,  24772608,   8429568,  1048576,   20736;
  10: 131072, 4718592, 39813120, 143130624, 263577600, 262766592, 139087872, 35389440, 3359232, 51200;

%F Formula

The row polynomials satisfy (4*n^3-20*n^2+33*n-18+(12*n^4-78*n^3+188*n^2-199*n+78)*x+(4*n^5-28*n^4+77*n^3-104*n^2+69*n-18)*x^2)*P(n,x) = (16*n^3-80*n^2+132*n-72+(48*n^4-216*n^3+272*n^2-4*n-120)*x+(16*n^5+8*n^4-324*n^3+726*n^2-496*n+40)*x^2+(24*n^5-104*n^4+126*n^3-60*n+8)*x^3)*P(n-1,x) - ((8*n^2-24*n+18)*x+(24*n^3-56*n^2+8*n+34)*x^2+(8*n^4+32*n^3-146*n^2+126*n-6)*x^3+(16*n^4-32*n^3-4*n^2+28*n-4)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 8*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 80*n^2 + 132*n - 72 + (48*n^4 - 216*n^3 + 272*n^2 - 4*n - 120)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (24*n^5 - 104*n^4 + 126*n^3 - 60*n + 8)*x^3)*P[n - 1, x] - ((8*n^2 - 24*n + 18)*x + (24*n^3 - 56*n^2 + 8*n + 34)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^3 - 20*n^2 + 33*n - 18 + (12*n^4 - 78*n^3 + 188*n^2 - 199*n + 78)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; for(n = 3, N, my(i = n); v[i] = ((16*n^3 - 80*n^2 + 132*n - 72 + (48*n^4 - 216*n^3 + 272*n^2 - 4*n - 120)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (24*n^5 - 104*n^4 + 126*n^3 - 60*n + 8)*x^3)*v[i - 1] - ((8*n^2 - 24*n + 18)*x + (24*n^3 - 56*n^2 + 8*n + 34)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^3 - 20*n^2 + 33*n - 18 + (12*n^4 - 78*n^3 + 188*n^2 - 199*n + 78)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*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 basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    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 Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = binomial(2n, 2k).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (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