Real-rooted atlas

A038231 → Veronese 2 (j=1) → θ+1

nonnegative coefficients

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

Recurrence

\(\left(128 - 128\,n + 4\,n x - 8\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(128\,x - 512\,n - 64\,n x - 2\,n x^{2} - 16\,n^{2} x - n^{2} x^{2} + 16\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-1024\,x + 64\,x^{2} + 32\,n x^{2} - 2\,n x^{3} + 32\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
288
348250
425632288
5128032031603
661442560728776
728672179201008447604
813107211468810752128256640
958982468812896768230451377029
1026214403932160774144307202007358664
11115343362162688056770563379204400639180598
125033164811534336038928384324403270400288207918112
1321810380859978547225303449628114944915200748871099961415
1493952409630534533121574436864224919552102502401397763925802724232
Technical details
Node id
n-590cd6d12c5e06dd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c4ccf282bde09205

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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:      8;
  3:     48,      2;
  4:    256,     32;
  5:   1280,    320,     3;
  6:   6144,   2560,    72;
  7:  28672,  17920,  1008,    4;
  8: 131072, 114688, 10752,  128;
  9: 589824, 688128, 96768, 2304, 5;
%F Formula
The row polynomials satisfy -(128*n-128-(4*n^3-8*n^2+4*n)*x)*P(n,x) = -(512*n-(16*n^3-16*n^2-64*n+128)*x-(n^3-n^2-2*n)*x^2)*P(n-1,x) - (1024*x-(32*n^2+32*n+64)*x^2+(2*n^2+2*n)*x^3)*d/dx 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[(-(512*n - (16*n^3 - 16*n^2 - 64*n + 128)*x - (n^3 - n^2 - 2*n)*x^2)*P[n - 1, x] - (1024*x - (32*n^2 + 32*n + 64)*x^2 + (2*n^2 + 2*n)*x^3)*D[P[n - 1, x], x])/(-128*n + 128 + (4*n^3 - 8*n^2 + 4*n)*x)]];
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] = (-(512*n - (16*n^3 - 16*n^2 - 64*n + 128)*x - (n^3 - n^2 - 2*n)*x^2)*v[i - 1] - (1024*x - (32*n^2 + 32*n + 64)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(-128*n + 128 + (4*n^3 - 8*n^2 + 4*n)*x)); 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.