Real-rooted atlas

A131431 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-1 - 8\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(2 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141620
2756112175
3101204806401250
4132081248332833288125
516320256010240204801638450000
6194564560243207296011673677824296875
7226167392492801971204730886307843604481718750

All 15 rows

Technical details
Node id
n-c461466144e48481 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-03d7b8417365c509

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  4,   16;
  2:  7,   56,   112;
  3: 10,  120,   480,    640;
  4: 13,  208,  1248,   3328,   3328;
  5: 16,  320,  2560,  10240,  20480,   16384;
  6: 19,  456,  4560,  24320,  72960,  116736,   77824;
  7: 22,  616,  7392,  49280, 197120,  473088,  630784,   360448;
  8: 25,  800, 11200,  89600, 448000, 1433600, 2867200,  3276800,  1638400;
  9: 28, 1008, 16128, 150528, 903168, 3612672, 9633792, 16515072, 16515072, 7340032;
%F Formula
The row polynomials satisfy P(n,x) = -(1+8*x+16*x^2)*P(n-2,x) + (2+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 8*x + 16*x^2)*P[n - 2, x] + (2 + 8*x)*P[n - 1, x]];
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 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 8*x + 16*x^2)*v[i - 2] + (2 + 8*x)*v[i - 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.