A281260 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8\,n + 8\,n x + 4\,n^{2} - 16\,n^{2} x - 24\,n^{3} x + 8\,n^{3} x^{2} - 7\,n^{4} x + 12\,n^{4} x^{2} + 6\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(-4 + 12\,x - 12\,x^{2} + 4\,x^{3} + 42\,n x - 64\,n x^{2} + 22\,n x^{3} + 4\,n^{2} + 59\,n^{2} x - 84\,n^{2} x^{2} + 51\,n^{2} x^{3} - 10\,n^{3} x - 124\,n^{3} x^{2} + 60\,n^{3} x^{3} - 7\,n^{4} x - 33\,n^{4} x^{2} + 54\,n^{4} x^{3} + 4\,n^{5} x^{2} + 22\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(4\,x - 12\,x^{2} + 12\,x^{3} - 4\,x^{4} + 8\,n x - 34\,n x^{2} + 44\,n x^{3} - 18\,n x^{4} - 48\,n^{2} x^{2} + 81\,n^{2} x^{3} - 33\,n^{2} x^{4} - 14\,n^{3} x^{2} + 50\,n^{3} x^{3} - 36\,n^{3} x^{4} + 15\,n^{4} x^{3} - 15\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	8														8
2	0	8	27													35
3	0	8	72	64												144
4	0	8	135	320	125											588
5	0	8	216	960	1000	216										2400
6	0	8	315	2240	4375	2520	343									9801
7	0	8	432	4480	14000	15120	5488	512								40040
8	0	8	567	8064	36750	63504	43218	10752	729							163592
9	0	8	720	13440	84000	211680	230496	107520	19440	1000						668304
10	0	8	891	21120	173250	598752	950796	709632	240570	33000	1331					2729350
11	0	8	1080	31680	330000	1496880	3259872	3548160	1924560	495000	53240	1728				11142208
12	0	8	1287	45760	589875	3397680	9711702	14496768	11467170	4719000	951665	82368	2197			45465480
13	0	8	1512	64064	1001000	7135128	25897872	50738688	55042416	33033000	10658648	1729728	123032	2744		185427840
14	0	8	1755	87360	1626625	14054040	63126063	157048320	223609815	184041000	86601515	22486464	2998905	178360	3375	755863605

Technical details

Node id: n-0307da5bd86a145e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-064abb064fdd3693

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: 0, 8;
  2: 0, 8,  27;
  3: 0, 8,  72,    64;
  4: 0, 8, 135,   320,   125;
  5: 0, 8, 216,   960,  1000,    216;
  6: 0, 8, 315,  2240,  4375,   2520,    343;
  7: 0, 8, 432,  4480, 14000,  15120,   5488,    512;
  8: 0, 8, 567,  8064, 36750,  63504,  43218,  10752,   729;
  9: 0, 8, 720, 13440, 84000, 211680, 230496, 107520, 19440, 1000;

%F Formula

The row polynomials satisfy (4*n^2+8*n-(7*n^4+24*n^3+16*n^2-8*n)*x+(n^6+6*n^5+12*n^4+8*n^3)*x^2)*P(n,x) = (4*n^2-4-(7*n^4+10*n^3-59*n^2-42*n-12)*x+(n^6+4*n^5-33*n^4-124*n^3-84*n^2-64*n-12)*x^2+(3*n^6+22*n^5+54*n^4+60*n^3+51*n^2+22*n+4)*x^3)*P(n-1,x) + ((8*n+4)*x-(14*n^3+48*n^2+34*n+12)*x^2+(2*n^5+15*n^4+50*n^3+81*n^2+44*n+12)*x^3-(2*n^5+15*n^4+36*n^3+33*n^2+18*n+4)*x^4)*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[((4*n^2 - 4 - (7*n^4 + 10*n^3 - 59*n^2 - 42*n - 12)*x + (n^6 + 4*n^5 - 33*n^4 - 124*n^3 - 84*n^2 - 64*n - 12)*x^2 + (3*n^6 + 22*n^5 + 54*n^4 + 60*n^3 + 51*n^2 + 22*n + 4)*x^3)*P[n - 1, x] + ((8*n + 4)*x - (14*n^3 + 48*n^2 + 34*n + 12)*x^2 + (2*n^5 + 15*n^4 + 50*n^3 + 81*n^2 + 44*n + 12)*x^3 - (2*n^5 + 15*n^4 + 36*n^3 + 33*n^2 + 18*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 8*n - (7*n^4 + 24*n^3 + 16*n^2 - 8*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*n^3)*x^2)]];
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] = ((4*n^2 - 4 - (7*n^4 + 10*n^3 - 59*n^2 - 42*n - 12)*x + (n^6 + 4*n^5 - 33*n^4 - 124*n^3 - 84*n^2 - 64*n - 12)*x^2 + (3*n^6 + 22*n^5 + 54*n^4 + 60*n^3 + 51*n^2 + 22*n + 4)*x^3)*v[i - 1] + ((8*n + 4)*x - (14*n^3 + 48*n^2 + 34*n + 12)*x^2 + (2*n^5 + 15*n^4 + 50*n^3 + 81*n^2 + 44*n + 12)*x^3 - (2*n^5 + 15*n^4 + 36*n^3 + 33*n^2 + 18*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 8*n - (7*n^4 + 24*n^3 + 16*n^2 - 8*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*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 - 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 Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 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
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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
[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.