Real-rooted atlas

A135065 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\,n^{2} + n^{3} x\right)\,P_{n}(x) = \left(4 + 4\,x + x^{2} + 8\,n + 10\,n x + 3\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 3\,n^{2} x^{2} + 2\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18816
2367227135
312838428864864
4400160018008001254725
51152576086405760180021623328
63136188163528031360147003528343107163
781925734412902414336089600322566272512466560

All 15 rows

Technical details
Node id
n-ba747c7b819ac5dd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     8,      8;
  2:    36,     72,      27;
  3:   128,    384,     288,      64;
  4:   400,   1600,    1800,     800,     125;
  5:  1152,   5760,    8640,    5760,    1800,     216;
  6:  3136,  18816,   35280,   31360,   14700,    3528,    343;
  7:  8192,  57344,  129024,  143360,   89600,   32256,   6272,    512;
  8: 20736, 165888,  435456,  580608,  453600,  217728,  63504,  10368,   729;
  9: 51200, 460800, 1382400, 2150400, 2016000, 1209600, 470400, 115200, 16200, 1000;
%F Formula
The row polynomials satisfy (2*n^2+n^3*x)*P(n,x) = (4*n^2+8*n+4+(2*n^3+8*n^2+10*n+4)*x+(n^3+3*n^2+3*n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 8*n + 4 + (2*n^3 + 8*n^2 + 10*n + 4)*x + (n^3 + 3*n^2 + 3*n + 1)*x^2)*P[n - 1, x])/(2*n^2 + n^3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 8*n + 4 + (2*n^3 + 8*n^2 + 10*n + 4)*x + (n^3 + 3*n^2 + 3*n + 1)*x^2)*v[i - 1])/(2*n^2 + n^3*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. [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.