A133080 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 8\,n x - 8\,n x^{2} + 8\,n^{2} x^{2}\right)\,P_{n}(x) = \left(20 + 24\,x - 32\,x^{3} - 12\,n + 44\,n x + 56\,n x^{2} + 16\,n x^{3} - 36\,n^{2} x + 48\,n^{2} x^{3} - 24\,n^{3} x^{2} - 32\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3 + 8\,x - 16\,x^{3} - 16\,x^{4} + 12\,n x + 36\,n x^{2} + 24\,n x^{3} + 12\,n^{2} x^{2} + 40\,n^{2} x^{3} + 32\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	1	8	12													21
3	2	20	48	32												102
4	1	16	72	128	80											297
5	2	36	192	448	480	192										1350
6	1	24	180	640	1200	1152	448									3645
7	2	52	432	1760	4000	5184	3584	1024								16038
8	1	32	336	1792	5600	10752	12544	8192	2304							41553
9	2	68	768	4480	15680	34944	50176	45056	23040	5120						179334
10	1	40	540	3840	16800	48384	94080	122880	103680	51200	11264					452709
11	2	84	1200	9120	43200	137088	301056	460800	483840	332800	135168	24576				1928934
12	1	48	792	7040	39600	152064	413952	811008	1140480	1126400	743424	294912	53248			4782969
13	2	100	1728	16192	96800	399168	1182720	2568192	4105728	4787200	3964928	2211840	745472	114688		20194758
14	1	56	1092	11648	80080	384384	1345344	3514368	6918912	10250240	11275264	8945664	4845568	1605632	245760	49424013

Technical details

Node id: n-f47aaf22712644c6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-39eb53de90e86874

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,  4;
  2: 1,  8,  12;
  3: 2, 20,  48,   32;
  4: 1, 16,  72,  128,    80;
  5: 2, 36, 192,  448,   480,   192;
  6: 1, 24, 180,  640,  1200,  1152,   448;
  7: 2, 52, 432, 1760,  4000,  5184,  3584,  1024;
  8: 1, 32, 336, 1792,  5600, 10752, 12544,  8192,  2304;
  9: 2, 68, 768, 4480, 15680, 34944, 50176, 45056, 23040, 5120;

%F Formula

The row polynomials satisfy (4+8*n*x+(8*n^2-8*n)*x^2)*P(n,x) = -(12*n-20+(36*n^2-44*n-24)*x+(24*n^3-56*n)*x^2+(32*n^3-48*n^2-16*n+32)*x^3)*P(n-1,x) + (3+(12*n+8)*x+(12*n^2+36*n)*x^2+(40*n^2+24*n-16)*x^3+(32*n^2-16)*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[(-(12*n - 20 + (36*n^2 - 44*n - 24)*x + (24*n^3 - 56*n)*x^2 + (32*n^3 - 48*n^2 - 16*n + 32)*x^3)*P[n - 1, x] + (3 + (12*n + 8)*x + (12*n^2 + 36*n)*x^2 + (40*n^2 + 24*n - 16)*x^3 + (32*n^2 - 16)*x^4)*D[P[n - 1, x], x])/(4 + 8*n*x + (8*n^2 - 8*n)*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] = (-(12*n - 20 + (36*n^2 - 44*n - 24)*x + (24*n^3 - 56*n)*x^2 + (32*n^3 - 48*n^2 - 16*n + 32)*x^3)*v[i - 1] + (3 + (12*n + 8)*x + (12*n^2 + 36*n)*x^2 + (40*n^2 + 24*n - 16)*x^3 + (32*n^2 - 16)*x^4)*deriv(v[i - 1], x))/(4 + 8*n*x + (8*n^2 - 8*n)*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 basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.