Real-rooted atlas

A131431 → D+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,x - 2\,x^{2} + 12\,n - 10\,n x + 3\,n x^{2} - 11\,n^{2} + 6\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(x^{3} - n x + 2\,n x^{2} + 3\,n x^{3} - 2\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
21428749
30606010130
40015610413273
500032016016496
6000057022819817
700000924308221254
80000001400400251825
900000002016504282548
10000000002790620313441
110000000003740748344522
1200000000004884888375809
130000000000062401040407320
1400000000000078261204439073
Technical details
Node id
n-142648292d3beb81 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-43d1dff77ec354ab

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  8,  4;
  2: 14, 28,   7;
  3:  0, 60,  60,  10;
  4:  0,  0, 156, 104,  13;
  5:  0,  0,   0, 320, 160,  16;
  6:  0,  0,   0,   0, 570, 228,   19;
  7:  0,  0,   0,   0,   0, 924,  308,   22;
  8:  0,  0,   0,   0,   0,   0, 1400,  400,  25;
  9:  0,  0,   0,   0,   0,   0,    0, 2016, 504, 28;
%F Formula
The row polynomials satisfy (3*n^3-11*n^2+12*n-4+(6*n^2-10*n+4)*x+(3*n-2)*x^2)*P(n,x) = ((3*n^3-2*n^2-n)*x+(6*n^2+2*n)*x^2+(3*n+1)*x^3)*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^3 - 2*n^2 - n)*x + (6*n^2 + 2*n)*x^2 + (3*n + 1)*x^3)*P[n - 1, x])/(3*n^3 - 11*n^2 + 12*n - 4 + (6*n^2 - 10*n + 4)*x + (3*n - 2)*x^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] = (((3*n^3 - 2*n^2 - n)*x + (6*n^2 + 2*n)*x^2 + (3*n + 1)*x^3)*v[i - 1])/(3*n^3 - 11*n^2 + 12*n - 4 + (6*n^2 - 10*n + 4)*x + (3*n - 2)*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. [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.