A152198 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1024 + 448\,x + 16\,x^{2} - 24\,x^{3} + 640\,n x + 96\,n x^{2} - 12\,n x^{3} + 224\,n^{2} x^{2} + 44\,n^{2} x^{3} + n^{2} x^{4} + 24\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(-3200\,x - 1760\,x^{2} - 432\,x^{3} - 40\,x^{4} - 4\,x^{5} + 1024\,n + 3200\,n x - 608\,n x^{2} - 544\,n x^{3} - 124\,n x^{4} - 8\,n x^{5} + 1088\,n^{2} x + 2384\,n^{2} x^{2} + 436\,n^{2} x^{3} + 16\,n^{2} x^{4} - n^{2} x^{5} + 320\,n^{3} x^{2} + 540\,n^{3} x^{3} + 109\,n^{3} x^{4} + 7\,n^{3} x^{5} + 32\,n^{4} x^{3} + 42\,n^{4} x^{4} + 5\,n^{4} x^{5} + n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-1024 - 4352\,x - 4416\,x^{2} - 2112\,x^{3} - 592\,x^{4} - 96\,x^{5} - 8\,x^{6} - 1536\,n x - 5056\,n x^{2} - 4368\,n x^{3} - 1640\,n x^{4} - 312\,n x^{5} - 24\,n x^{6} - 704\,n^{2} x^{2} - 1712\,n^{2} x^{3} - 1148\,n^{2} x^{4} - 286\,n^{2} x^{5} - 26\,n^{2} x^{6} - 96\,n^{3} x^{3} - 184\,n^{3} x^{4} - 92\,n^{3} x^{5} - 12\,n^{3} x^{6} - 4\,n^{4} x^{4} - 6\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	4							6
3	2	4							6
4	4	16	9						29
5	4	16	9						29
6	8	48	54	16					126
7	8	48	54	16					126
8	16	128	216	128	25				513
9	16	128	216	128	25				513
10	32	320	720	640	250	36			1998
11	32	320	720	640	250	36			1998
12	64	768	2160	2560	1500	432	49		7533
13	64	768	2160	2560	1500	432	49		7533
14	128	1792	6048	8960	7000	3024	686	64	27702

Technical details

Node id: n-3a8c7055e07fb582 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f613749a36a48642

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,   4;
  3:  2,   4;
  4:  4,  16,   9;
  5:  4,  16,   9;
  6:  8,  48,  54,  16;
  7:  8,  48,  54,  16;
  8: 16, 128, 216, 128, 25;
  9: 16, 128, 216, 128, 25;

%F Formula

The row polynomials satisfy (1024+(640*n+448)*x+(224*n^2+96*n+16)*x^2+(24*n^3+44*n^2-12*n-24)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (1024*n+(1088*n^2+3200*n-3200)*x+(320*n^3+2384*n^2-608*n-1760)*x^2+(32*n^4+540*n^3+436*n^2-544*n-432)*x^3+(n^5+42*n^4+109*n^3+16*n^2-124*n-40)*x^4+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) - (1024+(1536*n+4352)*x+(704*n^2+5056*n+4416)*x^2+(96*n^3+1712*n^2+4368*n+2112)*x^3+(4*n^4+184*n^3+1148*n^2+1640*n+592)*x^4+(6*n^4+92*n^3+286*n^2+312*n+96)*x^5+(2*n^4+12*n^3+26*n^2+24*n+8)*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[((1024*n + (1088*n^2 + 3200*n - 3200)*x + (320*n^3 + 2384*n^2 - 608*n - 1760)*x^2 + (32*n^4 + 540*n^3 + 436*n^2 - 544*n - 432)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] - (1024 + (1536*n + 4352)*x + (704*n^2 + 5056*n + 4416)*x^2 + (96*n^3 + 1712*n^2 + 4368*n + 2112)*x^3 + (4*n^4 + 184*n^3 + 1148*n^2 + 1640*n + 592)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(1024 + (640*n + 448)*x + (224*n^2 + 96*n + 16)*x^2 + (24*n^3 + 44*n^2 - 12*n - 24)*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] = ((1024*n + (1088*n^2 + 3200*n - 3200)*x + (320*n^3 + 2384*n^2 - 608*n - 1760)*x^2 + (32*n^4 + 540*n^3 + 436*n^2 - 544*n - 432)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] - (1024 + (1536*n + 4352)*x + (704*n^2 + 5056*n + 4416)*x^2 + (96*n^3 + 1712*n^2 + 4368*n + 2112)*x^3 + (4*n^4 + 184*n^3 + 1148*n^2 + 1640*n + 592)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(1024 + (640*n + 448)*x + (224*n^2 + 96*n + 16)*x^2 + (24*n^3 + 44*n^2 - 12*n - 24)*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 shift_x_plus_one of Triangle read by rows, A007318 rows repeated.?
  - Affine, reversal and Möbius substitutions (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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, A007318 rows repeated.?
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    
    ↑ Gamma criterion (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A274496 → x ↦ x+1 → (θ+1)²

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.