A152198 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 + 56\,x + 4\,x^{2} - 12\,x^{3} + 80\,n x + 24\,n x^{2} - 6\,n x^{3} + 56\,n^{2} x^{2} + 22\,n^{2} x^{3} + n^{2} x^{4} + 12\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(-400\,x - 440\,x^{2} - 216\,x^{3} - 40\,x^{4} - 8\,x^{5} + 64\,n + 400\,n x - 152\,n x^{2} - 272\,n x^{3} - 124\,n x^{4} - 16\,n x^{5} + 136\,n^{2} x + 596\,n^{2} x^{2} + 218\,n^{2} x^{3} + 16\,n^{2} x^{4} - 2\,n^{2} x^{5} + 80\,n^{3} x^{2} + 270\,n^{3} x^{3} + 109\,n^{3} x^{4} + 14\,n^{3} x^{5} + 16\,n^{4} x^{3} + 42\,n^{4} x^{4} + 10\,n^{4} x^{5} + n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-32 - 272\,x - 552\,x^{2} - 528\,x^{3} - 296\,x^{4} - 96\,x^{5} - 16\,x^{6} - 96\,n x - 632\,n x^{2} - 1092\,n x^{3} - 820\,n x^{4} - 312\,n x^{5} - 48\,n x^{6} - 88\,n^{2} x^{2} - 428\,n^{2} x^{3} - 574\,n^{2} x^{4} - 286\,n^{2} x^{5} - 52\,n^{2} x^{6} - 24\,n^{3} x^{3} - 92\,n^{3} x^{4} - 92\,n^{3} x^{5} - 24\,n^{3} x^{6} - 2\,n^{4} x^{4} - 6\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	2	8			10
3	2	8			10
4	4	32	36		72
5	4	32	36		72
6	8	96	216	128	448
7	8	96	216	128	448

All 15 rows

Technical details

Node id: n-760706415e8d0c94 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-4f7c344a4c25ac22

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 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;
  2:  2,   8;
  3:  2,   8;
  4:  4,  32,  36;
  5:  4,  32,  36;
  6:  8,  96, 216,  128;
  7:  8,  96, 216,  128;
  8: 16, 256, 864, 1024, 400;
  9: 16, 256, 864, 1024, 400;

%F Formula

The row polynomials satisfy (64+(80*n+56)*x+(56*n^2+24*n+4)*x^2+(12*n^3+22*n^2-6*n-12)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (64*n+(136*n^2+400*n-400)*x+(80*n^3+596*n^2-152*n-440)*x^2+(16*n^4+270*n^3+218*n^2-272*n-216)*x^3+(n^5+42*n^4+109*n^3+16*n^2-124*n-40)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (32+(96*n+272)*x+(88*n^2+632*n+552)*x^2+(24*n^3+428*n^2+1092*n+528)*x^3+(2*n^4+92*n^3+574*n^2+820*n+296)*x^4+(6*n^4+92*n^3+286*n^2+312*n+96)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((64*n + (136*n^2 + 400*n - 400)*x + (80*n^3 + 596*n^2 - 152*n - 440)*x^2 + (16*n^4 + 270*n^3 + 218*n^2 - 272*n - 216)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (32 + (96*n + 272)*x + (88*n^2 + 632*n + 552)*x^2 + (24*n^3 + 428*n^2 + 1092*n + 528)*x^3 + (2*n^4 + 92*n^3 + 574*n^2 + 820*n + 296)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((64*n + (136*n^2 + 400*n - 400)*x + (80*n^3 + 596*n^2 - 152*n - 440)*x^2 + (16*n^4 + 270*n^3 + 218*n^2 - 272*n - 216)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (32 + (96*n + 272)*x + (88*n^2 + 632*n + 552)*x^2 + (24*n^3 + 428*n^2 + 1092*n + 528)*x^3 + (2*n^4 + 92*n^3 + 574*n^2 + 820*n + 296)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)); 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 basis_map_A013609 of Triangle read by rows, A007318 rows repeated.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, A007318 rows repeated.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.