Real-rooted atlas

A132774 → Veronese 2 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-3 + 16\,x\right)\,P_{n}(x) = \left(-294\,x + 51\,x^{2} - 16\,x^{3} + 175\,n x - 36\,n x^{2} - 33\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{3} x\right)\,P_{n-4}(x) + \left(-210\,x^{2} + 88\,x^{3} + 88\,n x^{2} - 16\,n x^{3} - 8\,n^{2} x^{2}\right)\,P_{n-4}'(x) + \left(-28\,x^{3} + 16\,x^{4} + 8\,n x^{3}\right)\,P_{n-4}''(x) + \left(60 + 2\,x + 32\,x^{2} - 67\,n + 20\,n x + 21\,n^{2} - 4\,n^{2} x - 2\,n^{3}\right)\,P_{n-2}(x) + \left(66\,x - 56\,x^{2} - 56\,n x + 16\,n x^{2} + 8\,n^{2} x\right)\,P_{n-2}'(x) + \left(12\,x^{2} - 16\,x^{3} - 8\,n x^{2}\right)\,P_{n-2}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 5 + 5\,x\)
\(P_{3}(x) = 6 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
25510
36612
4018927
50201030
600391352
700421456
8000681785
9000721890
10000010521126
11000011022132
120000015025175
130000015626182
1400000020329232
Technical details
Node id
n-672fbd583ff49e05 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ef69c06de8e75fc8

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 2;
  2: 5,  5;
  3: 6,  6;
  4: 0, 18,  9;
  5: 0, 20, 10;
  6: 0,  0, 39, 13;
  7: 0,  0, 42, 14;
  8: 0,  0,  0, 68, 17;
  9: 0,  0,  0, 72, 18;
%F Formula
The row polynomials satisfy -(3-16*x)*P(n,x) = ((2*n^3-33*n^2+175*n-294)*x+(4*n^2-36*n+51)*x^2-16*x^3)*P(n-4,x) - ((8*n^2-88*n+210)*x^2+(16*n-88)*x^3)*d/dx P(n-4,x) + ((8*n-28)*x^3+16*x^4)*d^2/dx^2 P(n-4,x) - (2*n^3-21*n^2+67*n-60+(4*n^2-20*n-2)*x-32*x^2)*P(n-2,x) + ((8*n^2-56*n+66)*x+(16*n-56)*x^2)*d/dx P(n-2,x) - ((8*n-12)*x^2+16*x^3)*d^2/dx^2 P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 5 + 5*x, P(3,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 5 + 5*x;
P[3, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n^3 - 33*n^2 + 175*n - 294)*x + (4*n^2 - 36*n + 51)*x^2 - 16*x^3)*P[n - 4, x] - ((8*n^2 - 88*n + 210)*x^2 + (16*n - 88)*x^3)*D[P[n - 4, x], x] + ((8*n - 28)*x^3 + 16*x^4)*D[P[n - 4, x], {x, 2}] - (2*n^3 - 21*n^2 + 67*n - 60 + (4*n^2 - 20*n - 2)*x - 32*x^2)*P[n - 2, x] + ((8*n^2 - 56*n + 66)*x + (16*n - 56)*x^2)*D[P[n - 2, x], x] - ((8*n - 12)*x^2 + 16*x^3)*D[P[n - 2, x], {x, 2}])/(-3 + 16*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 5 + 5*x; v[4] = 6 + 6*x; for(n = 4, N, my(i = n + 1); v[i] = (((2*n^3 - 33*n^2 + 175*n - 294)*x + (4*n^2 - 36*n + 51)*x^2 - 16*x^3)*v[i - 4] - ((8*n^2 - 88*n + 210)*x^2 + (16*n - 88)*x^3)*deriv(v[i - 4], x) + ((8*n - 28)*x^3 + 16*x^4)*deriv(deriv(v[i - 4], x), x) - (2*n^3 - 21*n^2 + 67*n - 60 + (4*n^2 - 20*n - 2)*x - 32*x^2)*v[i - 2] + ((8*n^2 - 56*n + 66)*x + (16*n - 56)*x^2)*deriv(v[i - 2], x) - ((8*n - 12)*x^2 + 16*x^3)*deriv(deriv(v[i - 2], x), x))/(-3 + 16*x)); 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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [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
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.