A375853 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 - x^{2} + n + 5\,n x - n x^{2} - 7\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{3} x + n^{3} x^{2} - 8\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(2 - 8\,x + 10\,x^{2} - 12\,x^{3} + n + 2\,n x + 7\,n x^{2} + 26\,n x^{3} - n^{2} x - 27\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 5\,n^{3} x^{2} - 25\,n^{3} x^{3} + 4\,n^{4} x^{2} + 12\,n^{4} x^{3} + 4\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 6\,x^{2} + 10\,x^{3} - 6\,x^{4} - 4\,n x^{2} - 6\,n x^{3} + 10\,n x^{4} + 4\,n^{2} x^{2} - 6\,n^{2} x^{3} + 2\,n^{2} x^{4} + 16\,n^{3} x^{3} - 16\,n^{3} x^{4} + 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	2								2
2	8	32							40
3	20	224	180						424
4	40	864	1944	640					3488
5	70	2464	10692	9856	1750				24832
6	112	5824	41184	73216	36400	4032			160768
7	168	12096	126360	366080	351000	108864	8232		972800
8	240	22848	330480	1414400	2210000	1321920	279888	15360	5595136

All 15 rows

Technical details

Node id: n-af8518402ee2134c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f6c2d99c4aa2ab10

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:   2;
   2:   8,    32;
   3:  20,   224,     180;
   4:  40,   864,    1944,      640;
   5:  70,  2464,   10692,     9856,     1750;
   6: 112,  5824,   41184,    73216,    36400,     4032;
   7: 168, 12096,  126360,   366080,   351000,   108864,     8232;
   8: 240, 22848,  330480,  1414400,  2210000,  1321920,   279888,    15360;
   9: 330, 40128,  767448,  4547840, 10497500, 10232640,  4178328,   642048,   26730;
  10: 440, 66528, 1625184, 12733952, 40698000, 58605120, 38997728, 11556864, 1347192, 44000;

%F Formula

The row polynomials satisfy (n-1+(2*n^3-7*n^2+5*n)*x+(4*n^5-8*n^4+n^3+5*n^2-n-1)*x^2)*P(n,x) = (n+2+(2*n^3-n^2+2*n-8)*x+(4*n^5+4*n^4-5*n^3-27*n^2+7*n+10)*x^2+(12*n^5+12*n^4-25*n^3-4*n^2+26*n-12)*x^3)*P(n-1,x) + (2*x+(4*n^2-4*n-6)*x^2+(8*n^4+16*n^3-6*n^2-6*n+10)*x^3-(8*n^4+16*n^3-2*n^2-10*n+6)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

%t Mathematica

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 2 + (2*n^3 - n^2 + 2*n - 8)*x + (4*n^5 + 4*n^4 - 5*n^3 - 27*n^2 + 7*n + 10)*x^2 + (12*n^5 + 12*n^4 - 25*n^3 - 4*n^2 + 26*n - 12)*x^3)*P[n - 1, x] + (2*x + (4*n^2 - 4*n - 6)*x^2 + (8*n^4 + 16*n^3 - 6*n^2 - 6*n + 10)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 10*n + 6)*x^4)*D[P[n - 1, x], x])/(n - 1 + (2*n^3 - 7*n^2 + 5*n)*x + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((n + 2 + (2*n^3 - n^2 + 2*n - 8)*x + (4*n^5 + 4*n^4 - 5*n^3 - 27*n^2 + 7*n + 10)*x^2 + (12*n^5 + 12*n^4 - 25*n^3 - 4*n^2 + 26*n - 12)*x^3)*v[i - 1] + (2*x + (4*n^2 - 4*n - 6)*x^2 + (8*n^4 + 16*n^3 - 6*n^2 - 6*n + 10)*x^3 - (8*n^4 + 16*n^3 - 2*n^2 - 10*n + 6)*x^4)*deriv(v[i - 1], x))/(n - 1 + (2*n^3 - 7*n^2 + 5*n)*x + (4*n^5 - 8*n^4 + n^3 + 5*n^2 - n - 1)*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.?
    
    ↑ Liu–Wang theorem (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.