Real-rooted atlas

A193685 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1156
21112537
311891125235
41262176716251540
513542521904651312510427
6145740559019981310311562573013
71561190122506470117089820181178125529032
8168180624150174951688506139809712889913906253967195
918126224401641645122630656906145110756708124571195312530785747
101953675754809006276427575272737306632483085654261507005519765625247126450
111110500512325518063871633447291549480311788400616252901650280378313968931488281252050937445
12112666551933253408702380111162712286721227283200342234850155803035864865931577193264171124414062517585497797
1311438671293150611525282324242727362064421079859215695180501363814400954950836026535994162750118399908911220703125155666739742
1411611110243188610512502167937770179757721012939143360620139650131933332451053769413211204320610448702637991301417226064847161035156251421428484337
Technical details
Node id
n-802eabe88630cab9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-73b18cc810b5ad72

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: 1,  5;
  2: 1, 11,   25;
  3: 1, 18,   91,   125;
  4: 1, 26,  217,   671,    625;
  5: 1, 35,  425,  2190,   4651,    3125;
  6: 1, 45,  740,  5590,  19981,   31031,   15625;
  7: 1, 56, 1190, 12250,  64701,  170898,  201811,    78125;
  8: 1, 68, 1806, 24150, 174951,  688506, 1398097,  1288991,  390625;
  9: 1, 81, 2622, 44016, 416451, 2263065, 6906145, 11075670, 8124571, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n+4)*x)*P(n-1,x) - x^2*d/dx 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[(1 + (n + 4)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];
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] = (1 + (n + 4)*x)*v[i - 1] - x^2*deriv(v[i - 1], 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

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.