Real-rooted atlas

A131431 → D+1 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
22128756
340906010200
46520823410413624
596400640480160161792
613368414251520855228194864
7176107827723850308013863082212672
82251600490084008750560021004002532000
928022688064164642116817640940830245042878848
103413100125552976045570468723255014880418562031190464
114084114187005049089760109956942485610022440561074834452608
124815328268628140016483523443223931617582491575325607326888371060864
135606760374401258402860004633205491204804803088801430004576093601040402457600
146458428508691878244734738608601162161118060890390351651621521562608117391204435636096

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2} - 9\,n^{2} - 18\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 10\,x - 24\,n - 24\,n x + 18\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 4\,x\)
Technical details
Node id
n-c84c6091c7207354 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-d229a49127c81b0a)
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:   8,    4;
  2:  21,   28,    7;
  3:  40,   90,   60,    10;
  4:  65,  208,  234,   104,    13;
  5:  96,  400,  640,   480,   160,    16;
  6: 133,  684, 1425,  1520,   855,   228,   19;
  7: 176, 1078, 2772,  3850,  3080,  1386,  308,   22;
  8: 225, 1600, 4900,  8400,  8750,  5600, 2100,  400,  25;
  9: 280, 2268, 8064, 16464, 21168, 17640, 9408, 3024, 504, 28;
%F Formula
The row polynomials satisfy (3*n-2)*P(n,x) = (2*n+6+(6*n-2)*x)*P(n-1,x) + (1-2*x-3*x^2)*d/dx 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 + 6 + (6*n - 2)*x)*P[n - 1, x] + (1 - 2*x - 3*x^2)*D[P[n - 1, x], x])/(3*n - 2)]];
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 + 6 + (6*n - 2)*x)*v[i - 1] + (1 - 2*x - 3*x^2)*deriv(v[i - 1], x))/(3*n - 2)); 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

Candidate equalities ?

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.