A185410 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(-96\,x - 15\,x^{2} + 64\,n x + 16\,n x^{2} + 152\,n^{2} x^{2} - 320\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(16\,x - 104\,x^{2} - 20\,x^{3} + 16\,n x^{2} + 8\,n x^{3} + 64\,n^{2} x^{2} + 224\,n^{2} x^{3} - 352\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(20\,x^{2} - 16\,x^{3} - 4\,x^{4} - 16\,n^{2} x^{2} - 32\,n^{2} x^{3} + 48\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(2 - 4\,x + 24\,n x - 8\,n^{2} - 48\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x - 4\,x^{2} - 8\,n x + 8\,n x^{2} + 16\,n^{2} x + 16\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	4							4
2	0	8	9						17
3	0	16	90	16					122
4	0	32	540	576	25				1173
5	0	64	2664	8256	2900	36			13920
6	0	128	11952	82688	87700	12888	49		195405
7	0	256	50976	687360	1606000	760320	53214	64	3158190

All 15 rows

Technical details

Node id: n-3c62c08218d24292 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-4cf187fd49e24446

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,    4;
  2: 0,    8,      9;
  3: 0,   16,     90,       16;
  4: 0,   32,    540,      576,        25;
  5: 0,   64,   2664,     8256,      2900,        36;
  6: 0,  128,  11952,    82688,     87700,     12888,        49;
  7: 0,  256,  50976,   687360,   1606000,    760320,     53214,       64;
  8: 0,  512, 211392,  5121024,  22514000,  23791680,   5812968,   209408,     81;
  9: 0, 1024, 862848, 35611648, 268129600, 529698240, 296437456, 40893952, 796392, 100;

%F Formula

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = ((64*n-96)*x+(144*n^4-320*n^3+152*n^2+16*n-15)*x^2)*P(n-2,x) + (16*x+(64*n^2+16*n-104)*x^2+(64*n^4-352*n^3+224*n^2+8*n-20)*x^3)*d/dx P(n-2,x) - ((16*n^2-20)*x^2-(64*n^3-32*n^2-16)*x^3+(64*n^3-48*n^2+4)*x^4)*d^2/dx^2 P(n-2,x) - (8*n^2-2-(32*n^3-48*n^2+24*n-4)*x)*P(n-1,x) + ((16*n^2-8*n-8)*x-(32*n^3-16*n^2-8*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(((64*n - 96)*x + (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*P[n - 2, x] + (16*x + (64*n^2 + 16*n - 104)*x^2 + (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*D[P[n - 2, x], x] - ((16*n^2 - 20)*x^2 - (64*n^3 - 32*n^2 - 16)*x^3 + (64*n^3 - 48*n^2 + 4)*x^4)*D[P[n - 2, x], {x, 2}] - (8*n^2 - 2 - (32*n^3 - 48*n^2 + 24*n - 4)*x)*P[n - 1, x] + ((16*n^2 - 8*n - 8)*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 4*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((64*n - 96)*x + (144*n^4 - 320*n^3 + 152*n^2 + 16*n - 15)*x^2)*v[i - 2] + (16*x + (64*n^2 + 16*n - 104)*x^2 + (64*n^4 - 352*n^3 + 224*n^2 + 8*n - 20)*x^3)*deriv(v[i - 2], x) - ((16*n^2 - 20)*x^2 - (64*n^3 - 32*n^2 - 16)*x^3 + (64*n^3 - 48*n^2 + 4)*x^4)*deriv(deriv(v[i - 2], x), x) - (8*n^2 - 2 - (32*n^3 - 48*n^2 + 24*n - 4)*x)*v[i - 1] + ((16*n^2 - 8*n - 8)*x - (32*n^3 - 16*n^2 - 8*n + 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 4*n + 1)); 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 A triangular decomposition of the double factorial numbers A001147.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A triangular decomposition of the double factorial numbers A001147.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (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.