A064861 → Veronese 2 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(32\right)\,P_{n}(x) = \left(-706 + 2280\,x - 2208\,x^{2} + 256\,x^{3}\right)\,P_{n-6}(x) + \left(81 - 549\,x + 1044\,x^{2} - 576\,x^{3}\right)\,P_{n-6}'(x) + \left(-888 + 768\,x - 384\,x^{2}\right)\,P_{n-4}(x) + \left(-162 - 36\,x + 576\,x^{2}\right)\,P_{n-4}'(x) + \left(168 + 192\,x\right)\,P_{n-2}(x) + \left(81 - 144\,x\right)\,P_{n-2}'(x)\)

\(P_{0}(x) = 1 + 4\,x\)

\(P_{1}(x) = 1 + 16\,x\)

\(P_{2}(x) = 1 + 26\,x + 12\,x^{2}\)

\(P_{3}(x) = 1 + 50\,x + 84\,x^{2}\)

\(P_{4}(x) = 1 + 66\,x + 198\,x^{2} + 32\,x^{3}\)

\(P_{5}(x) = 1 + 102\,x + 576\,x^{2} + 320\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	1	4				5
1	1	16				17
2	1	26	12			39
3	1	50	84			135
4	1	66	198	32		297
5	1	102	576	320		999
6	1	124	963	992	80	2160
7	1	172	2043	3872	1040	7128

All 15 rows

Technical details

Node id: n-5b7718cd36abc7db · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9f8d13479cf57a48

Observed on the stored rows ?

Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
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,   4;
  1: 1,  16;
  2: 1,  26,   12;
  3: 1,  50,   84;
  4: 1,  66,  198,    32;
  5: 1, 102,  576,   320;
  6: 1, 124,  963,   992,    80;
  7: 1, 172, 2043,  3872,  1040;
  8: 1, 200, 2955,  7880,  4000,  192;
  9: 1, 260, 5295, 21344, 19600, 3072;

%F Formula

The row polynomials satisfy 32*P(n,x) = -(706-2280*x+2208*x^2-256*x^3)*P(n-6,x) + (81-549*x+1044*x^2-576*x^3)*d/dx P(n-6,x) - (888-768*x+384*x^2)*P(n-4,x) - (162+36*x-576*x^2)*d/dx P(n-4,x) + (168+192*x)*P(n-2,x) + (81-144*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 1 + 4*x, P(1,x) = 1 + 16*x, P(2,x) = 1 + 26*x + 12*x^2, P(3,x) = 1 + 50*x + 84*x^2, P(4,x) = 1 + 66*x + 198*x^2 + 32*x^3, P(5,x) = 1 + 102*x + 576*x^2 + 320*x^3.

%t Mathematica

P[0, x_] := 1 + 4*x;
P[1, x_] := 1 + 16*x;
P[2, x_] := 1 + 26*x + 12*x^2;
P[3, x_] := 1 + 50*x + 84*x^2;
P[4, x_] := 1 + 66*x + 198*x^2 + 32*x^3;
P[5, x_] := 1 + 102*x + 576*x^2 + 320*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(706 - 2280*x + 2208*x^2 - 256*x^3)*P[n - 6, x] + (81 - 549*x + 1044*x^2 - 576*x^3)*D[P[n - 6, x], x] - (888 - 768*x + 384*x^2)*P[n - 4, x] - (162 + 36*x - 576*x^2)*D[P[n - 4, x], x] + (168 + 192*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] = 1 + 4*x; v[2] = 1 + 16*x; v[3] = 1 + 26*x + 12*x^2; v[4] = 1 + 50*x + 84*x^2; v[5] = 1 + 66*x + 198*x^2 + 32*x^3; v[6] = 1 + 102*x + 576*x^2 + 320*x^3; for(n = 6, N, my(i = n + 1); v[i] = (-(706 - 2280*x + 2208*x^2 - 256*x^3)*v[i - 6] + (81 - 549*x + 1044*x^2 - 576*x^3)*deriv(v[i - 6], x) - (888 - 768*x + 384*x^2)*v[i - 4] - (162 + 36*x - 576*x^2)*deriv(v[i - 4], x) + (168 + 192*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 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 veronese_section_2_0 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.