Real-rooted atlas

A132440 → D+1 → (1/n)D²+(2/n)D+1/n → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + x - n\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2415
3138122
4343912186
57313678161304
6136365340130201992
722981610956801952413040
835816032856255511902732818864
95292864641276165110190436432124832
1074847611288819236171369198285646836167328
111021748023805429604809034272153304080585401177664
12135411231411408728511814010579862832240905610715441458240
13175316248673861645602618552835362115961077123613574808584811159168
1422242278910561229200653482068082361432839296417503252195972410145212883584
152773311361595234928561022021149749615885871228656687687272272730731237611835617069696
Technical details
Node id
n-fc2a1f4caddbe7bf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-528339e5ff5f0c6c

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   4,    1;
   3:  13,    8,     1;
   4:  34,   39,    12,     1;
   5:  73,  136,    78,    16,     1;
   6: 136,  365,   340,   130,    20,    1;
   7: 229,  816,  1095,   680,   195,   24,    1;
   8: 358, 1603,  2856,  2555,  1190,  273,   28,   1;
   9: 529, 2864,  6412,  7616,  5110, 1904,  364,  32,  1;
  10: 748, 4761, 12888, 19236, 17136, 9198, 2856, 468, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-6-x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 6 - x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(n - 6 - x)*v[i - 1] + (1 + x)*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, v[i])))};
T(10)

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. [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.