Real-rooted atlas

A038231 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-136 + 296\,x - 16\,x^{2} + 148\,n - 312\,n x + 16\,n x^{2} - 16\,n^{2} + 16\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(-512 + 1216\,x + 6\,x^{2} - 4\,x^{3} + 528\,n - 1277\,n x - 10\,n x^{2} + 4\,n x^{3} - 32\,n^{2} + 60\,n^{2} x + 4\,n^{2} x^{2} + 16\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-1024\,x - 2048\,x^{2} + 260\,x^{3} - 8\,x^{4} - 32\,n x + 130\,n x^{2} - 8\,n x^{3} + 32\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
144
217118
3761288
4352981451
51664680202364
679364320243112500
7378882598423242866224
8180224150528192644521350469
98519688478721451525520361850548
103997696466944010214405664072519745942
1118612224252313606825984515328107804451195720
128598323213408665643794432429158413200010621268288967
1339426457670202163227177779233386496140940818616521402878572
1417951621123625975808164003840024600576013581568264992146317321030104
Technical details
Node id
n-600060a71ca6ce1b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-57430bfb4303b064

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      4;
  2:     17,      1;
  3:     76,     12;
  4:    352,     98,      1;
  5:   1664,    680,     20;
  6:   7936,   4320,    243,    1;
  7:  37888,  25984,   2324,   28;
  8: 180224, 150528,  19264,  452,  1;
  9: 851968, 847872, 145152, 5520, 36;
%F Formula
The row polynomials satisfy (4*n^3-16*n^2+148*n-136+(16*n^2-312*n+296)*x+(16*n-16)*x^2)*P(n,x) = (16*n^3-32*n^2+528*n-512+(n^3+60*n^2-1277*n+1216)*x+(4*n^2-10*n+6)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((32*n^2-32*n-1024)*x-(2*n^2-130*n+2048)*x^2-(8*n-260)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 32*n^2 + 528*n - 512 + (n^3 + 60*n^2 - 1277*n + 1216)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((32*n^2 - 32*n - 1024)*x - (2*n^2 - 130*n + 2048)*x^2 - (8*n - 260)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 16*n^2 + 148*n - 136 + (16*n^2 - 312*n + 296)*x + (16*n - 16)*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; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^3 - 32*n^2 + 528*n - 512 + (n^3 + 60*n^2 - 1277*n + 1216)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((32*n^2 - 32*n - 1024)*x - (2*n^2 - 130*n + 2048)*x^2 - (8*n - 260)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 16*n^2 + 148*n - 136 + (16*n^2 - 312*n + 296)*x + (16*n - 16)*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

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.