A172431 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2\,x + 4\,x^{2} + 12\,n x - 36\,n x^{2} + 4\,n^{2} - 98\,n^{2} x + 132\,n^{2} x^{2} - 12\,n^{3} + 204\,n^{3} x - 252\,n^{3} x^{2} + 8\,n^{4} - 168\,n^{4} x + 264\,n^{4} x^{2} + 48\,n^{5} x - 144\,n^{5} x^{2} + 32\,n^{6} x^{2}\right)\,P_{n}(x) = \left(16\,x^{2} - 16\,x^{3} - 2\,n x + 80\,n x^{2} - 40\,n x^{3} - 30\,n^{2} x + 48\,n^{2} x^{2} + 84\,n^{2} x^{3} - 4\,n^{3} - 28\,n^{3} x + 8\,n^{3} x^{2} + 84\,n^{3} x^{3} + 8\,n^{4} - 24\,n^{4} x - 312\,n^{4} x^{2} - 72\,n^{4} x^{3} + 48\,n^{5} x + 128\,n^{5} x^{2} - 144\,n^{5} x^{3} + 32\,n^{6} x^{2} + 96\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + 2\,x^{3} - 8\,x^{4} + 7\,n x^{2} + 22\,n x^{3} - 24\,n x^{4} + 2\,n^{2} x^{2} + 16\,n^{2} x^{3} + 32\,n^{2} x^{4} - 4\,n^{3} x - 4\,n^{3} x^{2} + 60\,n^{3} x^{3} + 48\,n^{3} x^{4} - 24\,n^{4} x^{2} - 96\,n^{4} x^{3} - 16\,n^{5} x^{3} - 64\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	4	24													28
3	6	80	108												194
4	8	168	540	320											1036
5	10	288	1512	2240	750										4800
6	12	440	3240	8064	7000	1512									20268
7	14	624	5940	21120	31500	18144	2744								80086
8	16	840	9828	45760	99000	99792	41160	4608							301004
9	18	1088	15120	87360	250250	370656	271656	84480	7290						1087918
10	20	1368	22032	152320	546000	1081080	1177176	658944	160380	11000					3810320
11	22	1680	30780	248064	1071000	2673216	3923920	3294720	1459458	286000	15972				13004832
12	24	2024	41580	383040	1938000	5860512	10915632	12446720	8339760	3003000	484484	22464			43437240
13	26	2400	54648	566720	3291750	11721024	26589360	38697984	35443980	19448000	5813808	786240	30758		142446698
14	28	2808	70200	809600	5313000	21804552	58496592	104186880	122442840	92378000	42357744	10692864	1230320	41160	459826588

Technical details

Node id: n-83ca5953d5765abe · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-8b36a23aa5145d0b

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  2;
  2:  4,   24;
  3:  6,   80,   108;
  4:  8,  168,   540,   320;
  5: 10,  288,  1512,  2240,    750;
  6: 12,  440,  3240,  8064,   7000,   1512;
  7: 14,  624,  5940, 21120,  31500,  18144,   2744;
  8: 16,  840,  9828, 45760,  99000,  99792,  41160,  4608;
  9: 18, 1088, 15120, 87360, 250250, 370656, 271656, 84480, 7290;

%F Formula

The row polynomials satisfy (8*n^4-12*n^3+4*n^2+(48*n^5-168*n^4+204*n^3-98*n^2+12*n+2)*x+(32*n^6-144*n^5+264*n^4-252*n^3+132*n^2-36*n+4)*x^2)*P(n,x) = (8*n^4-4*n^3+(48*n^5-24*n^4-28*n^3-30*n^2-2*n)*x+(32*n^6+128*n^5-312*n^4+8*n^3+48*n^2+80*n+16)*x^2+(96*n^6-144*n^5-72*n^4+84*n^3+84*n^2-40*n-16)*x^3)*P(n-1,x) - (4*n^3*x+(24*n^4+4*n^3-2*n^2-7*n-1)*x^2+(16*n^5+96*n^4-60*n^3-16*n^2-22*n-2)*x^3+(64*n^5-48*n^3-32*n^2+24*n+8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^4 - 4*n^3 + (48*n^5 - 24*n^4 - 28*n^3 - 30*n^2 - 2*n)*x + (32*n^6 + 128*n^5 - 312*n^4 + 8*n^3 + 48*n^2 + 80*n + 16)*x^2 + (96*n^6 - 144*n^5 - 72*n^4 + 84*n^3 + 84*n^2 - 40*n - 16)*x^3)*P[n - 1, x] - (4*n^3*x + (24*n^4 + 4*n^3 - 2*n^2 - 7*n - 1)*x^2 + (16*n^5 + 96*n^4 - 60*n^3 - 16*n^2 - 22*n - 2)*x^3 + (64*n^5 - 48*n^3 - 32*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(8*n^4 - 12*n^3 + 4*n^2 + (48*n^5 - 168*n^4 + 204*n^3 - 98*n^2 + 12*n + 2)*x + (32*n^6 - 144*n^5 + 264*n^4 - 252*n^3 + 132*n^2 - 36*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^4 - 4*n^3 + (48*n^5 - 24*n^4 - 28*n^3 - 30*n^2 - 2*n)*x + (32*n^6 + 128*n^5 - 312*n^4 + 8*n^3 + 48*n^2 + 80*n + 16)*x^2 + (96*n^6 - 144*n^5 - 72*n^4 + 84*n^3 + 84*n^2 - 40*n - 16)*x^3)*v[i - 1] - (4*n^3*x + (24*n^4 + 4*n^3 - 2*n^2 - 7*n - 1)*x^2 + (16*n^5 + 96*n^4 - 60*n^3 - 16*n^2 - 22*n - 2)*x^3 + (64*n^5 - 48*n^3 - 32*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(8*n^4 - 12*n^3 + 4*n^2 + (48*n^5 - 168*n^4 + 204*n^3 - 98*n^2 + 12*n + 2)*x + (32*n^6 - 144*n^5 + 264*n^4 - 252*n^3 + 132*n^2 - 36*n + 4)*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 derivative of Even row Pascal-square read by antidiagonals.?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? 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 Even row Pascal-square read by antidiagonals.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Even row Pascal-square read by antidiagonals.?
    
    ↑ Three-term recurrence criterion (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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023