A008517 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(8\,x + 14\,x^{2} + 8\,n x - 6\,n x^{2} - 16\,n^{2} x - 80\,n^{2} x^{2} + 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 2\,x^{2} + 16\,x^{3} - 4\,n x + 8\,n x^{2} + 4\,n x^{3} + 16\,n^{2} x^{2} - 88\,n^{2} x^{3} + 32\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 2\,x^{3} + 3\,x^{4} - 2\,n x^{2} + 2\,n x^{4} + 16\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-1 + 2\,n - 8\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x^{2} + 4\,n x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	8							9
2	1	32	54						87
3	1	88	522	384					995
4	1	208	2952	7104	3000				13265
5	1	456	13068	70400	92700	25920			202545
6	1	960	50490	513920	1453500	1223424	246960		3489255
7	1	1976	179550	3132800	16100500	28270944	16715664	2580480	66981915

All 15 rows

Technical details

Node id: n-a6a9ef684f8dadd1 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-3c9a530453f07c86

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,    8;
  2: 1,   32,      54;
  3: 1,   88,     522,      384;
  4: 1,  208,    2952,     7104,       3000;
  5: 1,  456,   13068,    70400,      92700,      25920;
  6: 1,  960,   50490,   513920,    1453500,    1223424,      246960;
  7: 1, 1976,  179550,  3132800,   16100500,   28270944,    16715664,     2580480;
  8: 1, 4016,  605340, 17000000,  144137500,  447842304,   540288504,   238970880,   29393280;
  9: 1, 8104, 1969632, 85218560, 1119125000, 5592865824, 11706340296, 10379962368, 3591462240, 362880000;

%F Formula

The row polynomials satisfy (2*n-1)*P(n,x) = -((16*n^2-8*n-8)*x-(72*n^3-80*n^2-6*n+14)*x^2)*P(n-2,x) - ((4*n+2)*x-(16*n^2+8*n+2)*x^2-(32*n^3-88*n^2+4*n+16)*x^3)*d/dx P(n-2,x) - ((2*n+1)*x^2-(16*n^2-2)*x^3+(16*n^2-2*n-3)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-1+(16*n^2-8*n)*x)*P(n-1,x) + (4*n*x-(16*n^2-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((16*n^2 - 8*n - 8)*x - (72*n^3 - 80*n^2 - 6*n + 14)*x^2)*P[n - 2, x] - ((4*n + 2)*x - (16*n^2 + 8*n + 2)*x^2 - (32*n^3 - 88*n^2 + 4*n + 16)*x^3)*D[P[n - 2, x], x] - ((2*n + 1)*x^2 - (16*n^2 - 2)*x^3 + (16*n^2 - 2*n - 3)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 1 + (16*n^2 - 8*n)*x)*P[n - 1, x] + (4*n*x - (16*n^2 - 4)*x^2)*D[P[n - 1, x], x])/(2*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] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-((16*n^2 - 8*n - 8)*x - (72*n^3 - 80*n^2 - 6*n + 14)*x^2)*v[i - 2] - ((4*n + 2)*x - (16*n^2 + 8*n + 2)*x^2 - (32*n^3 - 88*n^2 + 4*n + 16)*x^3)*deriv(v[i - 2], x) - ((2*n + 1)*x^2 - (16*n^2 - 2)*x^3 + (16*n^2 - 2*n - 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 1 + (16*n^2 - 8*n)*x)*v[i - 1] + (4*n*x - (16*n^2 - 4)*x^2)*deriv(v[i - 1], x))/(2*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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 Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A201637 → (θ+1)²

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.