Real-rooted atlas

A079621 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
2246037121
31927208883612161
4192096001776014440436148081
523040144000355200433200261660627011279801
63225602419200745920012129600109897205266884104420539631369
7516096045158400167086080339628800410282880294945504116950960197481771398961761
8928972809289728004010065920978130944014770183680141573841928420469120284373748841778792155422807841
91857945600209018880001031159808002934392832005317266124806370822886405052281472002559363739207520182578097706781012434261023721
1040874803200510935040000283568947200092223774720001949664245760028031620700160277875480960001876866742080082722008358002149549182220250194150581117366299630361
1198099527680013488685056000831802245120003043384565760007353019441152001233391310807040146718253946880012387320497728007279536735504002837404920530406605125575338469596384117056161301265353169
1225505877196800382588157952000259522300477440010550399827968000286767758204928005497401271025664076293492052377600772968799058227205678038653693120029509011173516160103039958975279042171407184451960208919770666777349768597919934961
137141645615104001160517412454400085878288521625600384034553738035200115981626651770880025013175783166771203967261586723635200468934404761991168041336121398885913602685320016789970560125021816890005235239519610757025672076046796522706828672993347643526121347094823271661081
142142493684531200037493639479296000030057400982568960001466313750636134400048712283193743769600116728153654778265600208281233302990848000281360642857194700800289352849792201395200225566881410357527040131272907734505496960553274550598359408001596982726976843388028265720601028096202315126151113962211393695557886847095721
Technical details
Node id
n-4fca5dbd91c267d1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-59ff5d09747816f6

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:          4,           5;
  2:         24,          60,           37;
  3:        192,         720,          888,          361;
  4:       1920,        9600,        17760,        14440,         4361;
  5:      23040,      144000,       355200,       433200,       261660,        62701;
  6:     322560,     2419200,      7459200,     12129600,     10989720,      5266884,      1044205;
  7:    5160960,    45158400,    167086080,    339628800,    410282880,    294945504,    116950960,     19748177;
  8:   92897280,   928972800,   4010065920,   9781309440,  14770183680,  14157384192,   8420469120,   2843737488,   417787921;
  9: 1857945600, 20901888000, 103115980800, 293439283200, 531726612480, 637082288640, 505228147200, 255936373920, 75201825780, 9770678101;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+2+(2*n+7)*x-(2*n-7)*x^2-(2*n-2)*x^3)*P(n-1,x) + (2*x+6*x^2+6*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[((2*n + 2 + (2*n + 7)*x - (2*n - 7)*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*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] = ((2*n + 2 + (2*n + 7)*x - (2*n - 7)*x^2 - (2*n - 2)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*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

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  4. [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
  5. [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
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  7. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023