Real-rooted atlas

A216973 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 2\,x + x^{2} - 5\,n + 4\,n x - 2\,n x^{2} + 2\,n^{2} - 3\,n^{2} x + n^{2} x^{2} + n^{3} x\right)\,P_{n}(x) = \left(-n + n x^{2} + 2\,n^{2} + 2\,n^{2} x + n^{2} x^{2} + n^{2} x^{3} + n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
26410
31530954
428968416224
54522033018025800
666420900840330362592
791714199526601785546497840
8120112038646720644033608406422528
915316566804146161827013608579612248162208
101902340111602856044100428402604093601710100166400
1123131901732551480947101136529009046200143552310121433664
1227642242574087120186120266112260568174240772202112030361441105920
133255460368941401403410555662806606605456883153151229803003039001692768896
1437869165132421621659059011171161513512148948810630625405401881884149649141966823936
Technical details
Node id
n-2cbd808c0ff0a0d6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0bc6c5656007bb44

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:   0;
  1:   1;
  2:   6,    4;
  3:  15,   30,    9;
  4:  28,   96,   84,    16;
  5:  45,  220,  330,   180,    25;
  6:  66,  420,  900,   840,   330,    36;
  7:  91,  714, 1995,  2660,  1785,   546,   49;
  8: 120, 1120, 3864,  6720,  6440,  3360,  840,   64;
  9: 153, 1656, 6804, 14616, 18270, 13608, 5796, 1224, 81;
%F Formula
The row polynomials satisfy (2*n^2-5*n+3+(n^3-3*n^2+4*n-2)*x+(n^2-2*n+1)*x^2)*P(n,x) = (2*n^2-n+(n^3+2*n^2)*x+(n^3+n^2+n)*x^2+n^2*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - n + (n^3 + 2*n^2)*x + (n^3 + n^2 + n)*x^2 + n^2*x^3)*P[n - 1, x])/(2*n^2 - 5*n + 3 + (n^3 - 3*n^2 + 4*n - 2)*x + (n^2 - 2*n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - n + (n^3 + 2*n^2)*x + (n^3 + n^2 + n)*x^2 + n^2*x^3)*v[i - 1])/(2*n^2 - 5*n + 3 + (n^3 - 3*n^2 + 4*n - 2)*x + (n^2 - 2*n + 1)*x^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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.