Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → reverse → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(5 + 5\,x + x^{2} - 2\,n - 3\,n x - n x^{2}\right)\,P_{n-1}(x) + \left(4 + 8\,x + 5\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21115531
3491016815233
4257706715317522047
51539526871114737155920320417
610299421497100463068311768138877226711
7759053610997284118080675327242088264509341402764265
860944133017967752138103061628489401443864214391452646712114736622543
9528445132100422859242981330599701314154978587320237139185102547911640930115975522808721
104913492333058015599307543917545198362019457703158273318585564976831515930875711025107149436785707987414855
114870269293593604273119664697982374238280331193167865285011572111848465765985115994232727672207579519041734818914213597129864952697
1251209054414110427658215021817692533057069136748795797343550909217630838502073763421272308835085231044188241564369844598043572527856040276444372236349051263
1356878092067493279034840196228252560947416563604597766356328722910670120267578663130156665068973321574243716529405486367085244221930686658037792673856396222247119089932240624528690337
1466492002181961945852757932663955433449470102381010488126129683300201164160811628595159411457005124117347608290608658137899913162798980238557688869093939902043016335020321516534795310086142031382958545775738427080567
Technical details
Node id
n-85e32806de9dc4d3 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a11451714fced61e

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:       3,        2;
  2:      11,       15,        5;
  3:      49,      101,       68,        15;
  4:     257,      706,      715,       317,        52;
  5:    1539,     5268,     7111,      4737,      1559,      203;
  6:   10299,    42149,    71004,     63068,     31176,     8138,      877;
  7:   75905,   361099,   728411,    808067,    532724,   208826,    45093,     4140;
  8:  609441,  3301796,  7752138,  10306162,   8489401,  4438642,  1439145,   264671,   21147;
  9: 5284451, 32100422, 85924298, 133059970, 131415497, 85873202, 37139185, 10254791, 1640930, 115975;
%F Formula
The row polynomials satisfy P(n,x) = -(2*n-5+(3*n-5)*x+(n-1)*x^2)*P(n-1,x) + (4+8*x+5*x^2+x^3)*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[-(2*n - 5 + (3*n - 5)*x + (n - 1)*x^2)*P[n - 1, x] + (4 + 8*x + 5*x^2 + x^3)*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] = -(2*n - 5 + (3*n - 5)*x + (n - 1)*x^2)*v[i - 1] + (4 + 8*x + 5*x^2 + x^3)*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.