Real-rooted atlas

A075497 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 15\,x + 14\,x^{2} + 4\,x^{3} - 2\,n - 8\,n x - 10\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 10\,x + 18\,x^{2} + 14\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
211302061
349202272120643
42571412286025368327897
51539105362844437896249446496109855
61029984298284016504544498816260416561281698517
7759057221982913644646453685235846682432288595252992028798171
86094416603592310085528244929613583041614203654492105280338778885413632529934641
9528445164200844343697192106447976021026479522747942464237690784013126132484200780805937920010497231031
104913492366116031039723017561403615868832311323248506474619205476158515240340391424193820224005486050816694855680222342446317
11487026929718720854647865879192189939062424499090685840912037030752118301809017610894847261446982840849922967137489927524545638486294466565007982447027
1251209054418220855316460087270770026445655309367807327574960162909496418562464132720857627228555308800218191473121281236809573580847083966177281081049169920113231613952119390847880969
1356878092067986558069680784913010243637933250883672124261701259552291414438485600503444033002624648828585161472623914315277824442199476450304224570231377920773993960570881622926324121615638472458243000651104707279
146649200218191238917055158610655821733797656081904808390420180749328032165253145972115040102023332483279361502049386119782416848330237776896143307788214149129100195219445760418409745412096013169317265203202540225675509762265839280128079231493790070405
Technical details
Node id
n-d6aa0b915b7d3713 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f59c2fbf8eb76307

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:      11,       30,        20;
  3:      49,      202,       272,        120;
  4:     257,     1412,      2860,       2536,        832;
  5:    1539,    10536,     28444,      37896,      24944,       6496;
  6:   10299,    84298,    284016,     504544,     498816,     260416,      56128;
  7:   75905,   722198,   2913644,    6464536,    8523584,    6682432,    2885952,     529920;
  8:  609441,  6603592,  31008552,   82449296,  135830416,  142036544,   92105280,   33877888,   5413632;
  9: 5284451, 64200844, 343697192, 1064479760, 2102647952, 2747942464, 2376907840, 1312613248, 420078080, 59379200;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(2*n-5+(8*n-15)*x+(10*n-14)*x^2+(4*n-4)*x^3)*P(n-1,x) + (2+10*x+18*x^2+14*x^3+4*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 - 5 + (8*n - 15)*x + (10*n - 14)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*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 - 5 + (8*n - 15)*x + (10*n - 14)*x^2 + (4*n - 4)*x^3)*v[i - 1] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*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. [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.