A081577 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n x + 48\,n x^{2} + 32\,n x^{3} + 24\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{3} x^{3}\right)\,P_{n}(x) = \left(12 + 120\,x + 400\,x^{2} + 512\,x^{3} + 256\,x^{4} - 12\,n - 60\,n x + 32\,n x^{2} + 432\,n x^{3} + 384\,n x^{4} - 36\,n^{2} x - 168\,n^{2} x^{2} - 96\,n^{2} x^{3} + 128\,n^{2} x^{4} - 24\,n^{3} x^{2} - 80\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3 + 42\,x + 220\,x^{2} + 520\,x^{3} + 512\,x^{4} + 128\,x^{5} + 12\,n x + 132\,n x^{2} + 496\,n x^{3} + 688\,n x^{4} + 192\,n x^{5} + 12\,n^{2} x^{2} + 96\,n^{2} x^{3} + 208\,n^{2} x^{4} + 64\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	8							10
2	6	48	48						102
3	16	192	480	256					944
4	44	704	2784	3584	1280				8396
5	120	2400	13056	27648	23040	6144			72408
6	328	7872	54528	161792	222720	135168	28672		611080
7	896	25088	211200	808960	1576960	1572864	745472	131072	5072512

All 15 rows

Technical details

Node id: n-a5faf66d342ecf2f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6dd3d0485dde69b3

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:    2,      8;
  2:    6,     48,      48;
  3:   16,    192,     480,      256;
  4:   44,    704,    2784,     3584,     1280;
  5:  120,   2400,   13056,    27648,    23040,     6144;
  6:  328,   7872,   54528,   161792,   222720,   135168,    28672;
  7:  896,  25088,  211200,   808960,  1576960,  1572864,   745472,   131072;
  8: 2448,  78336,  776448,  3649536,  9262080, 13123584, 10149888,  3932160,   589824;
  9: 6688, 240768, 2746368, 15310848, 48015360, 88989696, 97714176, 61341696, 20054016, 2621440;

%F Formula

The row polynomials satisfy (12*n*x+(24*n^2+48*n)*x^2+(16*n^3+48*n^2+32*n)*x^3)*P(n,x) = -(12*n-12+(36*n^2+60*n-120)*x+(24*n^3+168*n^2-32*n-400)*x^2+(80*n^3+96*n^2-432*n-512)*x^3-(128*n^2+384*n+256)*x^4)*P(n-1,x) + (3+(12*n+42)*x+(12*n^2+132*n+220)*x^2+(96*n^2+496*n+520)*x^3+(208*n^2+688*n+512)*x^4+(64*n^2+192*n+128)*x^5)*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[(-(12*n - 12 + (36*n^2 + 60*n - 120)*x + (24*n^3 + 168*n^2 - 32*n - 400)*x^2 + (80*n^3 + 96*n^2 - 432*n - 512)*x^3 - (128*n^2 + 384*n + 256)*x^4)*P[n - 1, x] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 220)*x^2 + (96*n^2 + 496*n + 520)*x^3 + (208*n^2 + 688*n + 512)*x^4 + (64*n^2 + 192*n + 128)*x^5)*D[P[n - 1, x], x])/(12*n*x + (24*n^2 + 48*n)*x^2 + (16*n^3 + 48*n^2 + 32*n)*x^3)]];
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] = (-(12*n - 12 + (36*n^2 + 60*n - 120)*x + (24*n^3 + 168*n^2 - 32*n - 400)*x^2 + (80*n^3 + 96*n^2 - 432*n - 512)*x^3 - (128*n^2 + 384*n + 256)*x^4)*v[i - 1] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 220)*x^2 + (96*n^2 + 496*n + 520)*x^3 + (208*n^2 + 688*n + 512)*x^4 + (64*n^2 + 192*n + 128)*x^5)*deriv(v[i - 1], x))/(12*n*x + (24*n^2 + 48*n)*x^2 + (16*n^3 + 48*n^2 + 32*n)*x^3)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013611 of Pascal-(1,2,1) array read by antidiagonals.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,2,1) array read by antidiagonals.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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 Pascal-(1,2,1) array read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,2,1) array read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,2,1) array read by antidiagonals.?
    
    ↑ Liu–Wang theorem with a lag row (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
[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.