Real-rooted atlas

A131431 → θ+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 6\,x + 2\,x^{2} + 3\,n + 9\,n x + 3\,n x^{2} - 9\,n^{2} - 27\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 4\,x^{2} + 9\,n x + 6\,n x^{2} - 27\,n^{2} x - 18\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(2\,x^{2} + 3\,n x^{2} - 9\,n^{2} x^{2}\right)\,P_{n-2}''(x) + \left(-10 - 10\,x - 24\,n - 24\,n x + 18\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-10\,x - 24\,n x + 18\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18816
2216321105
34028024040600
4659751625650653380
59629768640624014409619488
613383794003346550186202793133116641
717622352170016299376184800468164928176728640

All 15 rows

Technical details
Node id
n-2a2bde98ff098755 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-4732d0911d5a78c5

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:   8,      8;
  2:  21,     63,      21;
  3:  40,    280,     240,      40;
  4:  65,    975,    1625,     650,       65;
  5:  96,   2976,    8640,    6240,     1440,      96;
  6: 133,   8379,   40033,   46550,    18620,    2793,     133;
  7: 176,  22352,  170016,  299376,   184800,   46816,    4928,    176;
  8: 225,  57375,  680625, 1748250,  1563975,  595350,  103950,   8100,   225;
  9: 280, 143080, 2612400, 9549400, 11907000, 6391560, 1646400, 210000, 12600, 280;
%F Formula
The row polynomials satisfy (9*n^2-21*n+10)*P(n,x) = -(9*n^2-3*n-2+(27*n^2-9*n-6)*x+(9*n^2-3*n-2)*x^2)*P(n-2,x) - ((27*n^2-9*n-6)*x+(18*n^2-6*n-4)*x^2)*d/dx P(n-2,x) - (9*n^2-3*n-2)*x^2*d^2/dx^2 P(n-2,x) + (18*n^2-24*n-10+(18*n^2-24*n-10)*x)*P(n-1,x) + (18*n^2-24*n-10)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 3*n - 2 + (27*n^2 - 9*n - 6)*x + (9*n^2 - 3*n - 2)*x^2)*P[n - 2, x] - ((27*n^2 - 9*n - 6)*x + (18*n^2 - 6*n - 4)*x^2)*D[P[n - 2, x], x] - (9*n^2 - 3*n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (18*n^2 - 24*n - 10 + (18*n^2 - 24*n - 10)*x)*P[n - 1, x] + (18*n^2 - 24*n - 10)*x*D[P[n - 1, x], x])/(9*n^2 - 21*n + 10)]];
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] = 8 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 3*n - 2 + (27*n^2 - 9*n - 6)*x + (9*n^2 - 3*n - 2)*x^2)*v[i - 2] - ((27*n^2 - 9*n - 6)*x + (18*n^2 - 6*n - 4)*x^2)*deriv(v[i - 2], x) - (9*n^2 - 3*n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (18*n^2 - 24*n - 10 + (18*n^2 - 24*n - 10)*x)*v[i - 1] + (18*n^2 - 24*n - 10)*x*deriv(v[i - 1], x))/(9*n^2 - 21*n + 10)); 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 Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.