A133080 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 24\,x + 8\,x^{2} - 16\,n + 16\,n x + 8\,n^{2}\right)\,P_{n}(x) = \left(6 - 124\,x - 56\,x^{2} + 48\,x^{3} - 102\,n + 48\,n x + 136\,n x^{2} - 32\,n x^{3} + 96\,n^{2} + 64\,n^{2} x - 64\,n^{2} x^{2} - 24\,n^{3} - 32\,n^{3} x\right)\,P_{n-1}(x) + \left(1 - 18\,x - 44\,x^{2} + 8\,x^{3} + 32\,x^{4} - 16\,n - 24\,n x + 48\,n x^{2} + 64\,n x^{3} + 12\,n^{2} + 40\,n^{2} x + 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	5	12	4						21
3	12	42	40	8					102
4	9	56	120	96	16				297
5	20	146	400	496	256	32			1350
6	13	132	540	1120	1200	576	64		3645
7	28	314	1464	3640	5120	3936	1408	128	16038

All 15 rows

Technical details

Node id: n-e6837f1248e981f4 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2ff653cb92aef93c

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:  4,   2;
  2:  5,  12,    4;
  3: 12,  42,   40,     8;
  4:  9,  56,  120,    96,    16;
  5: 20, 146,  400,   496,   256,    32;
  6: 13, 132,  540,  1120,  1200,   576,    64;
  7: 28, 314, 1464,  3640,  5120,  3936,  1408,   128;
  8: 17, 240, 1456,  4928, 10080, 12544,  8960,  3072,  256;
  9: 36, 546, 3616, 13664, 32256, 48832, 46592, 26112, 7168, 512;

%F Formula

The row polynomials satisfy (8*n^2-16*n+2+(16*n-24)*x+8*x^2)*P(n,x) = -(24*n^3-96*n^2+102*n-6+(32*n^3-64*n^2-48*n+124)*x+(64*n^2-136*n+56)*x^2+(32*n-48)*x^3)*P(n-1,x) + (12*n^2-16*n+1+(40*n^2-24*n-18)*x+(32*n^2+48*n-44)*x^2+(64*n+8)*x^3+32*x^4)*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[(-(24*n^3 - 96*n^2 + 102*n - 6 + (32*n^3 - 64*n^2 - 48*n + 124)*x + (64*n^2 - 136*n + 56)*x^2 + (32*n - 48)*x^3)*P[n - 1, x] + (12*n^2 - 16*n + 1 + (40*n^2 - 24*n - 18)*x + (32*n^2 + 48*n - 44)*x^2 + (64*n + 8)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(8*n^2 - 16*n + 2 + (16*n - 24)*x + 8*x^2)]];
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] = (-(24*n^3 - 96*n^2 + 102*n - 6 + (32*n^3 - 64*n^2 - 48*n + 124)*x + (64*n^2 - 136*n + 56)*x^2 + (32*n - 48)*x^3)*v[i - 1] + (12*n^2 - 16*n + 1 + (40*n^2 - 24*n - 18)*x + (32*n^2 + 48*n - 44)*x^2 + (64*n + 8)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(8*n^2 - 16*n + 2 + (16*n - 24)*x + 8*x^2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.