A081578 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(108\,n x + 324\,n x^{2} + 162\,n x^{3} + 162\,n^{2} x^{2} + 243\,n^{2} x^{3} + 81\,n^{3} x^{3}\right)\,P_{n}(x) = \left(192 + 1404\,x + 3348\,x^{2} + 2916\,x^{3} + 972\,x^{4} - 192\,n - 756\,n x + 54\,n x^{2} + 2349\,n x^{3} + 1458\,n x^{4} - 432\,n^{2} x - 1566\,n^{2} x^{2} - 810\,n^{2} x^{3} + 486\,n^{2} x^{4} - 216\,n^{3} x^{2} - 567\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(64 + 672\,x + 2628\,x^{2} + 4590\,x^{3} + 3240\,x^{4} + 486\,x^{5} + 192\,n x + 1584\,n x^{2} + 4428\,n x^{3} + 4455\,n x^{4} + 729\,n x^{5} + 144\,n^{2} x^{2} + 864\,n^{2} x^{3} + 1377\,n^{2} x^{4} + 243\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	6							8
2	7	42	27						76
3	20	180	324	108					632
4	61	732	2106	1836	405				5140
5	182	2730	10908	16308	8910	1458			40496
6	547	9846	50301	107028	100845	39366	5103		313036
7	1640	34440	214488	595080	813240	542376	163296	17496	2382056

All 15 rows

Technical details

Node id: n-f3cdfd035a79c74f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2d7feb89f1e92983

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,      6;
  2:     7,     42,      27;
  3:    20,    180,     324,      108;
  4:    61,    732,    2106,     1836,      405;
  5:   182,   2730,   10908,    16308,     8910,     1458;
  6:   547,   9846,   50301,   107028,   100845,    39366,     5103;
  7:  1640,  34440,  214488,   595080,   813240,   542376,   163296,    17496;
  8:  4921, 118104,  867780,  2972808,  5364630,  5219640,  2653560,   647352,   59049;
  9: 14762, 398574, 3375432, 13775832, 31011660, 40162068, 29760696, 12124728, 2480058, 196830;

%F Formula

The row polynomials satisfy (108*n*x+(162*n^2+324*n)*x^2+(81*n^3+243*n^2+162*n)*x^3)*P(n,x) = -(192*n-192+(432*n^2+756*n-1404)*x+(216*n^3+1566*n^2-54*n-3348)*x^2+(567*n^3+810*n^2-2349*n-2916)*x^3-(486*n^2+1458*n+972)*x^4)*P(n-1,x) + (64+(192*n+672)*x+(144*n^2+1584*n+2628)*x^2+(864*n^2+4428*n+4590)*x^3+(1377*n^2+4455*n+3240)*x^4+(243*n^2+729*n+486)*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[(-(192*n - 192 + (432*n^2 + 756*n - 1404)*x + (216*n^3 + 1566*n^2 - 54*n - 3348)*x^2 + (567*n^3 + 810*n^2 - 2349*n - 2916)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*P[n - 1, x] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2628)*x^2 + (864*n^2 + 4428*n + 4590)*x^3 + (1377*n^2 + 4455*n + 3240)*x^4 + (243*n^2 + 729*n + 486)*x^5)*D[P[n - 1, x], x])/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*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] = (-(192*n - 192 + (432*n^2 + 756*n - 1404)*x + (216*n^3 + 1566*n^2 - 54*n - 3348)*x^2 + (567*n^3 + 810*n^2 - 2349*n - 2916)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*v[i - 1] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2628)*x^2 + (864*n^2 + 4428*n + 4590)*x^3 + (1377*n^2 + 4455*n + 3240)*x^4 + (243*n^2 + 729*n + 486)*x^5)*deriv(v[i - 1], x))/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*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_A013610 of Pascal-(1,3,1) array.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,3,1) array.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,3,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,3,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Pascal-(1,3,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.