A060187 → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-96\,x + 192\,x^{2} - n - 16\,n x - 880\,n x^{2} + n^{2} + 64\,n^{2} x + 1328\,n^{2} x^{2} - 16\,n^{3} x - 784\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 96\,x^{2} + 288\,x^{3} + 4\,n x - 32\,n x^{2} - 1216\,n x^{3} - 4\,n^{2} x + 32\,n^{2} x^{2} + 1632\,n^{2} x^{3} + 16\,n^{3} x^{2} - 768\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{2} - 32\,x^{3} + 64\,x^{4} + 4\,n x^{2} + 48\,n x^{3} - 256\,n x^{4} - 4\,n^{2} x^{2} - 64\,n^{2} x^{3} + 320\,n^{2} x^{4} + 32\,n^{3} x^{3} - 128\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(1 - 32\,x - n + 80\,n x - 64\,n^{2} x + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,n x - 16\,n x^{2} + 4\,n^{2} x + 32\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
1	1				1
2	1				1
3	1	16			17
4	1	80			81
5	1	288	720		1009
6	1	928	8784		9713
7	1	2864	68976	62464	134305
8	1	8688	447984	1418240	1874913

All 15 rows

Technical details

Node id: n-a843ab95b98f04cf · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-98b8e10712046853

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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: 1;
   2: 1;
   3: 1,    16;
   4: 1,    80;
   5: 1,   288,      720;
   6: 1,   928,     8784;
   7: 1,  2864,    68976,     62464;
   8: 1,  8688,   447984,   1418240;
   9: 1, 26176,  2630880,  19484672,   8864000;
  10: 1, 78656, 14567904, 211226624, 323334400;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2-n-(16*n^3-64*n^2+16*n+96)*x+(144*n^4-784*n^3+1328*n^2-880*n+192)*x^2)*P(n-2,x) - ((4*n^2-4*n-6)*x-(16*n^3+32*n^2-32*n-96)*x^2-(64*n^4-768*n^3+1632*n^2-1216*n+288)*x^3)*d/dx P(n-2,x) - ((4*n^2-4*n-4)*x^2-(32*n^3-64*n^2+48*n-32)*x^3+(128*n^3-320*n^2+256*n-64)*x^4)*d^2/dx^2 P(n-2,x) - (n-1-(16*n^3-64*n^2+80*n-32)*x)*P(n-1,x) + ((4*n^2-6*n)*x-(16*n^3-32*n^2+16*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n - (16*n^3 - 64*n^2 + 16*n + 96)*x + (144*n^4 - 784*n^3 + 1328*n^2 - 880*n + 192)*x^2)*P[n - 2, x] - ((4*n^2 - 4*n - 6)*x - (16*n^3 + 32*n^2 - 32*n - 96)*x^2 - (64*n^4 - 768*n^3 + 1632*n^2 - 1216*n + 288)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4*n - 4)*x^2 - (32*n^3 - 64*n^2 + 48*n - 32)*x^3 + (128*n^3 - 320*n^2 + 256*n - 64)*x^4)*D[P[n - 2, x], {x, 2}] - (n - 1 - (16*n^3 - 64*n^2 + 80*n - 32)*x)*P[n - 1, x] + ((4*n^2 - 6*n)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 - n - (16*n^3 - 64*n^2 + 16*n + 96)*x + (144*n^4 - 784*n^3 + 1328*n^2 - 880*n + 192)*x^2)*v[i - 2] - ((4*n^2 - 4*n - 6)*x - (16*n^3 + 32*n^2 - 32*n - 96)*x^2 - (64*n^4 - 768*n^3 + 1632*n^2 - 1216*n + 288)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4*n - 4)*x^2 - (32*n^3 - 64*n^2 + 48*n - 32)*x^3 + (128*n^3 - 320*n^2 + 256*n - 64)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - 1 - (16*n^3 - 64*n^2 + 80*n - 32)*x)*v[i - 1] + ((4*n^2 - 6*n)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from gamma_polynomial of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) =…?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
    
    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: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

type_b_eulerian → γ → (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.