Real-rooted atlas

A136124 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
212312063
360227286120693
43601782330527228409009
52520153543739145494276566720135135
620160145512437314700471630694302664604802297295
71814401511208539094610677379126809569030916357096060480043648605
818144001710792070532748166074196244260729229799676135049676453285606652800916620705
919958400209959920981155268267320913246797583715458925184424314254821192102726171235207983360021082276215
102395008002778978240145033421764483352806890908819168126343961031121883737968805913763883495521577689806089601037836800527056905375
11311351040039479728320227449143648785882655348180948404213629151949617553353304048018275406022459615826970745206061209373441392204038401452971520014230536445125
1243589145600599418852480377648393587214414094329592371210164201246795499009661090674181098577888564038646146346866241222032226149240616110408997139202291722070400217945728000412685556908625
136538371840009688709116800662379606777602766351579178327874407555753321613261114171134244739255802441527836329555464422373732398017124149889083720476068123188355865621103022647872039936739046400348713164800012793252264167375
1410461394944000166134577996800122451542582976055522078582072321730184977380845639198670671518788665837198692999188614380077515812785301478612563498643357041615672683638085436941941614957425644156672422650167488064073471867015680059281238016000422177324717523375
Technical details
Node id
n-784df835ae395c37 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4123153381f31efd

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:        3,         4;
  2:       12,        31,        20;
  3:       60,       227,       286,        120;
  4:      360,      1782,      3305,       2722,        840;
  5:     2520,     15354,     37391,      45494,      27656,       6720;
  6:    20160,    145512,    437314,     700471,     630694,     302664,      60480;
  7:   181440,   1511208,   5390946,   10677379,   12680956,    9030916,    3570960,     604800;
  8:  1814400,  17107920,  70532748,  166074196,  244260729,  229799676,  135049676,   45328560,   6652800;
  9: 19958400, 209959920, 981155268, 2673209132, 4679758371, 5458925184, 4243142548, 2119210272, 617123520, 79833600;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (n+2+(2*n+5)*x+(n+3)*x^2)*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 + 2 + (2*n + 5)*x + (n + 3)*x^2)*P[n - 1, 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 + 2 + (2*n + 5)*x + (n + 3)*x^2)*v[i - 1])/(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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.