Real-rooted atlas

A135065 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 5\,x + x^{2} + 16\,n + 20\,n x + 4\,n x^{2}\right)\,P_{n-2}(x) + \left(5\,x + 2\,x^{2} + 20\,n x + 8\,n x^{2}\right)\,P_{n-2}'(x) + \left(x^{2} + 4\,n x^{2}\right)\,P_{n-2}''(x) + \left(16 + 8\,x - 8\,n - 4\,n x\right)\,P_{n-1}(x) + \left(8\,x - 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18412
23645990
312830414416592
440016251375350253775
5115275961026045007203624264
6313632585661994459012005132349159887
78192131776388416377664152320277762240641088448

All 15 rows

Technical details
Node id
n-3aee65bb256eaddb · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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,       4;
  2:    36,      45,        9;
  3:   128,     304,      144,       16;
  4:   400,    1625,     1375,      350,       25;
  5:  1152,    7596,    10260,     4500,      720,      36;
  6:  3136,   32585,    66199,    44590,    12005,    1323,      49;
  7:  8192,  131776,   388416,   377664,   152320,   27776,    2240,     64;
  8: 20736,  510705,  2133135,  2881494,  1634661,  438858,   57834,   3564,   81;
  9: 51200, 1917100, 11164500, 20420500, 15666000, 5810700, 1113000, 111000, 5400, 100;
%F Formula
The row polynomials satisfy P(n,x) = (16*n+4+(20*n+5)*x+(4*n+1)*x^2)*P(n-2,x) + ((20*n+5)*x+(8*n+2)*x^2)*d/dx P(n-2,x) + (4*n+1)*x^2*d^2/dx^2 P(n-2,x) - (8*n-16+(4*n-8)*x)*P(n-1,x) - (4*n-8)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 4*x;
P[n_, x_] := P[n, x] = Expand[(16*n + 4 + (20*n + 5)*x + (4*n + 1)*x^2)*P[n - 2, x] + ((20*n + 5)*x + (8*n + 2)*x^2)*D[P[n - 2, x], x] + (4*n + 1)*x^2*D[P[n - 2, x], {x, 2}] - (8*n - 16 + (4*n - 8)*x)*P[n - 1, x] - (4*n - 8)*x*D[P[n - 1, x], 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] = 8 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (16*n + 4 + (20*n + 5)*x + (4*n + 1)*x^2)*v[i - 2] + ((20*n + 5)*x + (8*n + 2)*x^2)*deriv(v[i - 2], x) + (4*n + 1)*x^2*deriv(deriv(v[i - 2], x), x) - (8*n - 16 + (4*n - 8)*x)*v[i - 1] - (4*n - 8)*x*deriv(v[i - 1], 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

  • 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.