Real-rooted atlas

A137597 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
210271855
33715120390481
4151825167214914944633
56744611124941677111159294648655
6326326796908711629641630768637518926552271
7170071628566629271487536198785915826036952671300666726121
8948281036735492194613256963221629812355653915549472582964395065487359761
95625956910597374684541177169152362273193140752872767263161558425075091441573537941203798199
103535027481760292935701971053551312246635594939361269774337848621326033239015997111244627888475995463817521951111
1123430840350701605237182698495688886822559165084247648888365630333549225925479389638798080265168543843394372646651513333618268381981009
12163254885266135318819774349046885634133002663203213015665597134228744385707769867272145828080385995674683897597491824530304644288805256346013807664311579013609
131192059223210177758321701412375388373860759562795975809197668880297728611796625928944155314127441341526731442221611068090468579575257060277820301143987554360669369674306255632272439773993727
149097183602172453747279151034893803481007178302822972904099665678980426689449156657057875748235699967490844270340968500747235255677392306152916136933275720005280943122321722119536445898431946794197428157261269599228878303
Technical details
Node id
n-8f823688897bdc90 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0e87d476abd07608

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,       4;
  2:     10,      27,       18;
  3:     37,     151,      203,        90;
  4:    151,     825,     1672,      1491,       494;
  5:    674,    4611,    12494,     16771,     11159,      2946;
  6:   3263,   26796,    90871,    162964,    163076,     86375,     18926;
  7:  17007,  162856,   662927,   1487536,   1987859,   1582603,    695267,    130066;
  8:  94828, 1036735,  4921946,  13256963,  22162981,  23556539,  15549472,   5829643,   950654;
  9: 562595, 6910597, 37468454, 117716915, 236227319, 314075287, 276726316, 155842507, 50914415, 7353794;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-4+(4*n-11)*x+(5*n-9)*x^2+(2*n-2)*x^3)*P(n-1,x) + (1+5*x+9*x^2+7*x^3+2*x^4)*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 - 4 + (4*n - 11)*x + (5*n - 9)*x^2 + (2*n - 2)*x^3)*P[n - 1, x] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*D[P[n - 1, x], x])/(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; for(n = 1, N, my(i = n + 1); v[i] = (-(n - 4 + (4*n - 11)*x + (5*n - 9)*x^2 + (2*n - 2)*x^3)*v[i - 1] + (1 + 5*x + 9*x^2 + 7*x^3 + 2*x^4)*deriv(v[i - 1], x))/(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.