A064861 → Veronese 2 (j=1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(32\right)\,P_{n}(x) = \left(-625 + 1731\,x - 1164\,x^{2} - 320\,x^{3}\right)\,P_{n-6}(x) + \left(81 - 549\,x + 1044\,x^{2} - 576\,x^{3}\right)\,P_{n-6}'(x) + \left(-1050 + 732\,x + 192\,x^{2}\right)\,P_{n-4}(x) + \left(-162 - 36\,x + 576\,x^{2}\right)\,P_{n-4}'(x) + \left(249 + 48\,x\right)\,P_{n-2}(x) + \left(81 - 144\,x\right)\,P_{n-2}'(x)\)

\(P_{0}(x) = 3\)

\(P_{1}(x) = 9 + 4\,x\)

\(P_{2}(x) = 18 + 12\,x\)

\(P_{3}(x) = 46 + 54\,x + 8\,x^{2}\)

\(P_{4}(x) = 72 + 135\,x + 36\,x^{2}\)

\(P_{5}(x) = 140 + 465\,x + 216\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	3					3
1	9	4				13
2	18	12				30
3	46	54	8			108
4	72	135	36			243
5	140	465	216	16		837
6	192	900	648	96		1836
7	318	2308	2778	720	32	6156

All 15 rows

Technical details

Node id: n-9a319b53d7728bfd · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b118b3db7c0b274c

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   3;
  1:   9,    4;
  2:  18,   12;
  3:  46,   54,     8;
  4:  72,  135,    36;
  5: 140,  465,   216,    16;
  6: 192,  900,   648,    96;
  7: 318, 2308,  2778,   720,   32;
  8: 405, 3756,  6363,  2520,  240;
  9: 607, 7896, 20153, 12860, 2160, 64;

%F Formula

The row polynomials satisfy 32*P(n,x) = -(625-1731*x+1164*x^2+320*x^3)*P(n-6,x) + (81-549*x+1044*x^2-576*x^3)*d/dx P(n-6,x) - (1050-732*x-192*x^2)*P(n-4,x) - (162+36*x-576*x^2)*d/dx P(n-4,x) + (249+48*x)*P(n-2,x) + (81-144*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 3, P(1,x) = 9 + 4*x, P(2,x) = 18 + 12*x, P(3,x) = 46 + 54*x + 8*x^2, P(4,x) = 72 + 135*x + 36*x^2, P(5,x) = 140 + 465*x + 216*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 3;
P[1, x_] := 9 + 4*x;
P[2, x_] := 18 + 12*x;
P[3, x_] := 46 + 54*x + 8*x^2;
P[4, x_] := 72 + 135*x + 36*x^2;
P[5, x_] := 140 + 465*x + 216*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(625 - 1731*x + 1164*x^2 + 320*x^3)*P[n - 6, x] + (81 - 549*x + 1044*x^2 - 576*x^3)*D[P[n - 6, x], x] - (1050 - 732*x - 192*x^2)*P[n - 4, x] - (162 + 36*x - 576*x^2)*D[P[n - 4, x], x] + (249 + 48*x)*P[n - 2, x] + (81 - 144*x)*D[P[n - 2, x], x])/(32)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 9 + 4*x; v[3] = 18 + 12*x; v[4] = 46 + 54*x + 8*x^2; v[5] = 72 + 135*x + 36*x^2; v[6] = 140 + 465*x + 216*x^2 + 16*x^3; for(n = 6, N, my(i = n + 1); v[i] = (-(625 - 1731*x + 1164*x^2 + 320*x^3)*v[i - 6] + (81 - 549*x + 1044*x^2 - 576*x^3)*deriv(v[i - 6], x) - (1050 - 732*x - 192*x^2)*v[i - 4] - (162 + 36*x - 576*x^2)*deriv(v[i - 4], x) + (249 + 48*x)*v[i - 2] + (81 - 144*x)*deriv(v[i - 2], x))/(32)); 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 veronese_section_2_1 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + …?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) f…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[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
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.