Real-rooted atlas

A347056 → reverse → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(3 + 9\,x + 7\,x^{2} - x^{3} - 2\,x^{4} + n + 6\,n x + 11\,n x^{2} + 8\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(-x - 5\,x^{2} - 9\,x^{3} - 7\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14711
210363278
320110200120450
4352607208804002295
556525196036403360123210773
6849524480112001568011648358447628
712015969072285605376060480376329984201204
81652520168006384015120022848021504011520026880820125
9220379529040129360369600702240887040718080337920704003247695
10286550047520242880813120186278429568003210240228096095744018022412557754
1136477227436042900016473604420416845644811531520109824006955520263577645260847593494
124551055611211272072031231209609600215255043536332842282240358758402050048070819841118208177324147
13560141051638001161160560560019459440499699209609600013837824014734720011275264058705920186368002723840651015225
146801848023296018054409609600371571201076275202372198403997593605125120004920115203429171201640038404816896065536002359598040
Technical details
Node id
n-19542de896724135 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-5969d25088d3741c

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,    7;
  2:  10,   36,    32;
  3:  20,  110,   200,    120;
  4:  35,  260,   720,    880,    400;
  5:  56,  525,  1960,   3640,   3360,   1232;
  6:  84,  952,  4480,  11200,  15680,  11648,   3584;
  7: 120, 1596,  9072,  28560,  53760,  60480,  37632,   9984;
  8: 165, 2520, 16800,  63840, 151200, 228480, 215040, 115200,  26880;
  9: 220, 3795, 29040, 129360, 369600, 702240, 887040, 718080, 337920, 70400;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n+3+(6*n+9)*x+(11*n+7)*x^2+(8*n-1)*x^3+(2*n-2)*x^4)*P(n-1,x) - (x+5*x^2+9*x^3+7*x^4+2*x^5)*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[Together[((n + 3 + (6*n + 9)*x + (11*n + 7)*x^2 + (8*n - 1)*x^3 + (2*n - 2)*x^4)*P[n - 1, x] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
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] = ((n + 3 + (6*n + 9)*x + (11*n + 7)*x^2 + (8*n - 1)*x^3 + (2*n - 2)*x^4)*v[i - 1] - (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.