Real-rooted atlas

A048993 → reverse → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
2123
316310
411221438
512075605161
61301953601556747
71424201400150537873753
8156798420085055796889820253
917213861058434755466202117520409116642
1019022502352011413525515023873574640459910713130
11111034654752031993510769221727110116600025650910230114607813
12113251158910079513537643769265564110352005503509865260225171231345921
13115672931573001797510113976724000796874570496675765092532530028778754914013223767233
141182101012642643763760308107801463862403946342405709273574007503501143091959467640106483141671430607
Technical details
Node id
n-662c588715f174cb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9e6242213e8ef38e

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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: 1;
  2: 1,  2;
  3: 1,  6,    3;
  4: 1, 12,   21,     4;
  5: 1, 20,   75,    60,     5;
  6: 1, 30,  195,   360,   155,     6;
  7: 1, 42,  420,  1400,  1505,   378,     7;
  8: 1, 56,  798,  4200,  8505,  5796,   889,    8;
  9: 1, 72, 1386, 10584, 34755, 46620, 21175, 2040, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) + (n*x^2+(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n-1+(2*n^2-4*n+2)*x)*P(n-1,x) - (n-1)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] + (n*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n - 1 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] - (n - 1)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] + (n*x^2 + (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n - 1 + (2*n^2 - 4*n + 2)*x)*v[i - 1] - (n - 1)*x^2*deriv(v[i - 1], x))/(n - 1)); 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.