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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
9	4862	127296	668304	1279488	1071000	411264	69972	4608	81							3636875
10	16796	503880	3139560	7441920	8139600	4395384	1173060	145920	7290	100						24963510
11	58786	1989680	14389650	40930560	55959750	39817008	15054270	2956800	280665	11000	121					171448290
12	208012	7845024	64721448	215738160	356919750	319800096	160367592	44884224	6762690	506000	15972	144				1177769112
13	742900	30904640	286833690	1099841600	2148128125	2344399200	1489127640	555709440	119884050	14300000	865150	22464	169			8090759068
14	2674440	121687020	1256239530	5459213760	12346083750	16000524540	12444852420	5910727680	1707565860	292792500	28342314	1415232	30758	196		55572150000
15	9694845	479039400	5449073175	26512449600	68352409125	103114491480	95693372775	55894924800	20633087475	4754950200	663864201	53202240	2229955	41160	225	381612830656

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.