Real-rooted atlas

A124932 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 32\,x - 24\,x^{2} - 8\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(32 + 48\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-24 - 24\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 3\,x\)
\(P_{2}(x) = 39 + 33\,x + 6\,x^{2}\)
\(P_{3}(x) = 152 + 204\,x + 84\,x^{2} + 10\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18311
23933678
31522048410450
4520960630170152295
516323840348015003002110773
648161377615960952030454832847628
71356845696645124928021840554472836201204
836864142848237888221760126000443529324104445820125
99728042624081792090048062496028224082320147601440553247695
10250624122496026611203379200277200015301445728801425602227519256612557754
116328323412992827904011911680112780807362432337075210771202336403234025087847593494
121570816926515224820224399027204283136032288256174894726836544190476036608045474319891177324147
13384204824600576721244161281280001537536001314593288225817638054016129729603203200552552622444004105651015225
14927744064081920204072960396963840526606080503543040357477120191093760771170402330328051651608080808326549351202359598040
Technical details
Node id
n-934a6db57f241fe8 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fa65b6b4f6cdb1f1

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:     8,      3;
  2:    39,     33,      6;
  3:   152,    204,     84,     10;
  4:   520,    960,    630,    170,     15;
  5:  1632,   3840,   3480,   1500,    300,     21;
  6:  4816,  13776,  15960,   9520,   3045,    483,    28;
  7: 13568,  45696,  64512,  49280,  21840,   5544,   728,    36;
  8: 36864, 142848, 237888, 221760, 126000,  44352,  9324,  1044,   45;
  9: 97280, 426240, 817920, 900480, 624960, 282240, 82320, 14760, 1440, 55;
%F Formula
The row polynomials satisfy P(n,x) = -(16+32*x+24*x^2+8*x^3+x^4)*P(n-4,x) + (32+48*x+24*x^2+4*x^3)*P(n-3,x) - (24+24*x+6*x^2)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8 + 3*x, P(2,x) = 39 + 33*x + 6*x^2, P(3,x) = 152 + 204*x + 84*x^2 + 10*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 3*x;
P[2, x_] := 39 + 33*x + 6*x^2;
P[3, x_] := 152 + 204*x + 84*x^2 + 10*x^3;
P[n_, x_] := P[n, x] = Expand[-(16 + 32*x + 24*x^2 + 8*x^3 + x^4)*P[n - 4, x] + (32 + 48*x + 24*x^2 + 4*x^3)*P[n - 3, x] - (24 + 24*x + 6*x^2)*P[n - 2, x] + (8 + 4*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] = 8 + 3*x; v[3] = 39 + 33*x + 6*x^2; v[4] = 152 + 204*x + 84*x^2 + 10*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(16 + 32*x + 24*x^2 + 8*x^3 + x^4)*v[i - 4] + (32 + 48*x + 24*x^2 + 4*x^3)*v[i - 3] - (24 + 24*x + 6*x^2)*v[i - 2] + (8 + 4*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

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.