A121574 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-n + 4\,n x + 8\,n x^{2} - 3\,n^{2} x + 6\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8\,x + 28\,x^{2} + 32\,x^{3} - n - 8\,n x + 2\,n x^{2} + 24\,n x^{3} - 3\,n^{2} x - 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(x + 12\,x^{2} + 36\,x^{3} + 16\,x^{4} + 3\,n x^{2} + 24\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	10	12													23
3	1	16	48	32												97
4	1	22	111	176	80											390
5	1	28	201	536	560	192										1518
6	1	34	318	1220	2120	1632	448									5773
7	1	40	462	2336	5840	7392	4480	1024								21575
8	1	46	633	3992	13205	23952	23632	11776	2304							79541
9	1	52	831	6296	26105	62652	88144	70912	29952	5120						290065
10	1	58	1056	9356	46835	141150	262276	299392	202752	74240	11264					1048380
11	1	64	1308	13280	78095	284880	664720	999616	956160	558080	180224	24576				3761004
12	1	70	1587	18176	122990	528510	1494115	2818720	3542112	2908160	1489664	430080	53248			13407433
13	1	76	1893	24152	185030	917400	3059665	6993520	11008800	11841920	8501504	3876864	1011712	114688		47537225
14	1	82	2226	31316	268130	1509060	5816860	15694600	29915280	40241600	37752704	24050688	9877504	2351104	245760	167756915

Technical details

Node id: n-609743fe373f06f1 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-552b265c5cde19a5

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: 1,  4;
  2: 1, 10,  12;
  3: 1, 16,  48,   32;
  4: 1, 22, 111,  176,    80;
  5: 1, 28, 201,  536,   560,   192;
  6: 1, 34, 318, 1220,  2120,  1632,   448;
  7: 1, 40, 462, 2336,  5840,  7392,  4480,  1024;
  8: 1, 46, 633, 3992, 13205, 23952, 23632, 11776,  2304;
  9: 1, 52, 831, 6296, 26105, 62652, 88144, 70912, 29952, 5120;

%F Formula

The row polynomials satisfy -(n+(3*n^2-4*n)*x-(6*n^2+8*n)*x^2)*P(n,x) = -(n+(3*n^2+8*n-8)*x+(12*n^2-2*n-28)*x^2-(24*n+32)*x^3)*P(n-1,x) + (x+(3*n+12)*x^2+(24*n+36)*x^3+(12*n+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[(-(n + (3*n^2 + 8*n - 8)*x + (12*n^2 - 2*n - 28)*x^2 - (24*n + 32)*x^3)*P[n - 1, x] + (x + (3*n + 12)*x^2 + (24*n + 36)*x^3 + (12*n + 16)*x^4)*D[P[n - 1, x], x])/(-n - (3*n^2 - 4*n)*x + (6*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] = (-(n + (3*n^2 + 8*n - 8)*x + (12*n^2 - 2*n - 28)*x^2 - (24*n + 32)*x^3)*v[i - 1] + (x + (3*n + 12)*x^2 + (24*n + 36)*x^3 + (12*n + 16)*x^4)*deriv(v[i - 1], x))/(-n - (3*n^2 - 4*n)*x + (6*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 reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? 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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.