Real-rooted atlas

A131431 → 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

\(\left(-2 + 3\,n - 2\,n x + 3\,n^{2} x\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} + 3\,n + 7\,n x + 4\,n x^{2} + 3\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
27282156
310609040200
41310423420865624
516160480640400961792
619228855152014256841334864
7223081386308038502772107817612672
82540021005600875084004900160022532000
928504302494081764021168164648064226828078848
103162041851488032550468724557029760125553100341190464
113474856102244056100942481099568976050490187004114408452608
123788873263256091575175824239316234432164835814002686253284811060864
134010409360457601430003088804804805491204633202860001258403744067605602457600
144312041173962608215215516516903903118060811621618608604734731878245086984286455636096
Technical details
Node id
n-6d32db1a89f2c14d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e99c3824c4864868

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:  4,   8;
  2:  7,  28,   21;
  3: 10,  60,   90,   40;
  4: 13, 104,  234,  208,    65;
  5: 16, 160,  480,  640,   400,    96;
  6: 19, 228,  855, 1520,  1425,   684,   133;
  7: 22, 308, 1386, 3080,  3850,  2772,  1078,  176;
  8: 25, 400, 2100, 5600,  8750,  8400,  4900, 1600,  225;
  9: 28, 504, 3024, 9408, 17640, 21168, 16464, 8064, 2268, 280;
%F Formula
The row polynomials satisfy (3*n-2+(3*n^2-2*n)*x)*P(n,x) = (3*n+1+(3*n^2+7*n+2)*x+(3*n^2+4*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[((3*n + 1 + (3*n^2 + 7*n + 2)*x + (3*n^2 + 4*n + 1)*x^2)*P[n - 1, x])/(3*n - 2 + (3*n^2 - 2*n)*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] = ((3*n + 1 + (3*n^2 + 7*n + 2)*x + (3*n^2 + 4*n + 1)*x^2)*v[i - 1])/(3*n - 2 + (3*n^2 - 2*n)*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.