Real-rooted atlas

A135065 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112416
272549135
332038414416864
4120020001200300254725
540328640720028805403623328
612544329283528019600588088249107163
7368641146881505281075204480010752134464466560

All 15 rows

Technical details
Node id
n-381980da7221376f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-49d880ced4b36b3a

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:     12,       4;
  2:     72,      54,       9;
  3:    320,     384,     144,      16;
  4:   1200,    2000,    1200,     300,      25;
  5:   4032,    8640,    7200,    2880,     540,     36;
  6:  12544,   32928,   35280,   19600,    5880,    882,     49;
  7:  36864,  114688,  150528,  107520,   44800,  10752,   1344,    64;
  8: 103680,  373248,  580608,  508032,  272160,  90720,  18144,  1944,   81;
  9: 281600, 1152000, 2073600, 2150400, 1411200, 604800, 168000, 28800, 2700, 100;
%F Formula
The row polynomials satisfy (n^3+n^2+n^2*x)*P(n,x) = (2*n^3+8*n^2+10*n+4+(n^3+6*n^2+9*n+4)*x+(n^2+2*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[((2*n^3 + 8*n^2 + 10*n + 4 + (n^3 + 6*n^2 + 9*n + 4)*x + (n^2 + 2*n + 1)*x^2)*P[n - 1, x])/(n^3 + n^2 + n^2*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] = ((2*n^3 + 8*n^2 + 10*n + 4 + (n^3 + 6*n^2 + 9*n + 4)*x + (n^2 + 2*n + 1)*x^2)*v[i - 1])/(n^3 + n^2 + n^2*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.