Real-rooted atlas

A126351 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + x - 5\,x^{2} - 9\,x^{3} - 4\,x^{4} + 2\,n + 10\,n x + 18\,n x^{2} + 14\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 7\,x - 19\,x^{2} - 25\,x^{3} - 16\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
210332770
337182296159674
41519872401257410257138
56745499178222866122853722182730
6326331938129436277868333059211195553071042066
717007194291945842254242440734623888155204606245763114164874
8948281239137704716722775086457314315839910246296406208218204065569206470546
9562595827946853894906203615635491882561787688723835878016566597582222528839385660213209494346
10353502757874781424503349183658750051890435391000158139913314651739120850808847155186808249455195738875708352951354066
11234308404225161153449044634168206820615444341829012277181355519677845155022411824247117771101243193408443532292831895244146851583923377096586
121632548853215822672289230955321570312469205730956873661480884429962277762197727838095573868123791001489812266886820067312614286001273593132950504663628118516957740768274
1311920592232547253386525032688213214977972385966086827569792177384678168463813201826478861211330367006733607994041116482025349041241035060791886176217249720754599985276361551163837685326932420504778
149097183602209627964005223524616419114616433789540654735289348702125018572818885152685778030609480972319717281330051356464297141545245372080111246635251543216468645656319682545474311851616133209302724768255005140596598148526790738
Technical details
Node id
n-bb7184253a86be79 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9daba0cbde89f7fd

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,       5;
  2:     10,      33,       27;
  3:     37,     182,      296,       159;
  4:    151,     987,     2401,      2574,      1025;
  5:    674,    5499,    17822,     28661,     22853,      7221;
  6:   3263,   31938,   129436,    277868,    333059,    211195,     55307;
  7:  17007,  194291,   945842,   2542424,   4073462,   3888155,   2046062,    457631;
  8:  94828, 1239137,  7047167,  22775086,  45731431,  58399102,  46296406,  20821820,   4065569;
  9: 562595, 8279468, 53894906, 203615635, 491882561, 787688723, 835878016, 566597582, 222528839, 38566021;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2*n+1+(10*n+1)*x+(18*n-5)*x^2+(14*n-9)*x^3+(4*n-4)*x^4)*P(n-1,x) - (1+7*x+19*x^2+25*x^3+16*x^4+4*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[((2*n + 1 + (10*n + 1)*x + (18*n - 5)*x^2 + (14*n - 9)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 4*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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] = ((2*n + 1 + (10*n + 1)*x + (18*n - 5)*x^2 + (14*n - 9)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 4*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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

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.