A120986 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2\,n - 2\,n x + 10\,n^{2} x - 6\,n^{3} + 10\,n^{3} x + 2\,n^{3} x^{2} + 4\,n^{4} - 30\,n^{4} x + 12\,n^{5} x - 6\,n^{5} x^{2} + 4\,n^{6} x^{2}\right)\,P_{n}(x) = \left(16\,x + 28\,x^{2} + 4\,x^{3} - 4\,n - 84\,n x - 102\,n x^{2} - 14\,n x^{3} + 20\,n^{2} + 110\,n^{2} x + 62\,n^{2} x^{2} + 6\,n^{2} x^{3} - 28\,n^{3} + 23\,n^{3} x + 131\,n^{3} x^{2} + 24\,n^{3} x^{3} + 12\,n^{4} - 108\,n^{4} x - 153\,n^{4} x^{2} - 24\,n^{4} x^{3} + 43\,n^{5} x + 19\,n^{5} x^{2} - 8\,n^{5} x^{3} + 15\,n^{6} x^{2} + 12\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} + 6\,x^{3} + 2\,x^{4} + 10\,n x + 16\,n x^{2} - 20\,n x^{3} - 6\,n x^{4} - n^{2} - 9\,n^{2} x + 3\,n^{2} x^{2} + 7\,n^{2} x^{3} + n^{3} - 2\,n^{3} x - 38\,n^{3} x^{2} + 27\,n^{3} x^{3} + 12\,n^{3} x^{4} + 3\,n^{4} x + 18\,n^{4} x^{2} - 15\,n^{4} x^{3} - 6\,n^{4} x^{4} + n^{5} x^{2} + 7\,n^{5} x^{3} - 8\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	4							6
3	5	24	9						38
4	14	112	108	16					250
5	42	480	810	320	25				1677
6	132	1980	4950	3520	750	36			11368
7	429	8008	27027	29120	11375	1512	49		77520
8	1430	32032	137592	203840	122500	30240	2744	64	530442

All 15 rows

Technical details

Node id: n-be7ed4e751810e40 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-0f300410a98ebdc3

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,      4;
   3:     5,     24,       9;
   4:    14,    112,     108,      16;
   5:    42,    480,     810,     320,      25;
   6:   132,   1980,    4950,    3520,     750,      36;
   7:   429,   8008,   27027,   29120,   11375,    1512,      49;
   8:  1430,  32032,  137592,  203840,  122500,   30240,    2744,     64;
   9:  4862, 127296,  668304, 1279488, 1071000,  411264,   69972,   4608,   81;
  10: 16796, 503880, 3139560, 7441920, 8139600, 4395384, 1173060, 145920, 7290, 100;

%F Formula

The row polynomials satisfy (4*n^4-6*n^3+2*n+(12*n^5-30*n^4+10*n^3+10*n^2-2*n)*x+(4*n^6-6*n^5+2*n^3)*x^2)*P(n,x) = (12*n^4-28*n^3+20*n^2-4*n+(43*n^5-108*n^4+23*n^3+110*n^2-84*n+16)*x+(15*n^6+19*n^5-153*n^4+131*n^3+62*n^2-102*n+28)*x^2+(12*n^6-8*n^5-24*n^4+24*n^3+6*n^2-14*n+4)*x^3)*P(n-1,x) + (n^3-n^2+(3*n^4-2*n^3-9*n^2+10*n-2)*x+(n^5+18*n^4-38*n^3+3*n^2+16*n-6)*x^2+(7*n^5-15*n^4+27*n^3+7*n^2-20*n+6)*x^3-(8*n^5+6*n^4-12*n^3+6*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^4 - 28*n^3 + 20*n^2 - 4*n + (43*n^5 - 108*n^4 + 23*n^3 + 110*n^2 - 84*n + 16)*x + (15*n^6 + 19*n^5 - 153*n^4 + 131*n^3 + 62*n^2 - 102*n + 28)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*P[n - 1, x] + (n^3 - n^2 + (3*n^4 - 2*n^3 - 9*n^2 + 10*n - 2)*x + (n^5 + 18*n^4 - 38*n^3 + 3*n^2 + 16*n - 6)*x^2 + (7*n^5 - 15*n^4 + 27*n^3 + 7*n^2 - 20*n + 6)*x^3 - (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 6*n^3 + 2*n + (12*n^5 - 30*n^4 + 10*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((12*n^4 - 28*n^3 + 20*n^2 - 4*n + (43*n^5 - 108*n^4 + 23*n^3 + 110*n^2 - 84*n + 16)*x + (15*n^6 + 19*n^5 - 153*n^4 + 131*n^3 + 62*n^2 - 102*n + 28)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*v[i - 1] + (n^3 - n^2 + (3*n^4 - 2*n^3 - 9*n^2 + 10*n - 2)*x + (n^5 + 18*n^4 - 38*n^3 + 3*n^2 + 16*n - 6)*x^2 + (7*n^5 - 15*n^4 + 27*n^3 + 7*n^2 - 20*n + 6)*x^3 - (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 6*n^3 + 2*n + (12*n^5 - 30*n^4 + 10*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*n^3)*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
  - 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

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.