Real-rooted atlas

A204420 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 3\,x + x^{2} - 6\,n - 10\,n x - 4\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
206915
30120330225675
40504019740256201102561425
503628801829520344358028690208930259398025
60399168002451556806001657207322653804453841701080560252170943775
706227020800449902252801350571622402156614059601932327797409212666463018261468225705556726875
8013076743680001084902698880038484804758400756767567556008910270008640062823975013800245634078690004108830350625306917176190625
903556874280960003328853233305600136032356940480003170618641270080046106517125869200428388015287232002483737571935620082164267295110001187451971330625172180535842940625
1001216451004088320001266870967333171200585402262458773760015754283838695001600272143621520322552003129532280526211560023959189847888895600117763105761803502003372238623548544750428670161650355625121042916697587259375
11051090942171709440000585731295560226816000301737959892077829120091984231896950929536001837767127671804288960025145569287256313095200238640970932339291316001551205316491475004360066096931859859232332001667198700878057597250189043541287806830625104217951276622630321875
120258520167388849766400003234071439623090626560001836645932405273801011200625059594181480875056640014165067533368033874592000224454461647355391147696002537723928213495776795040020473206263767612032678000115503808763487799290636004340130238030850534571300977604144656122725241500100004033341249813400625107865579571304422383140625
130155112100433309859840000002102017968391885479936000001304117024419607444766720000489826127284218137597184000012405662983655075793859200000223204349071963446404817600002925474742224068660750124000028144698309594915824575800000197259824406640254778619100009823066190536497776407530000329914388556295521925196250067100501041482659131631250062502520838281133375390625132135334974847917419347265625
14010888869450418352160768000000158869333502698649473843200000106872726103833287771357184000043892807243371051531671705600001227960788242781343037262976000024712673619936384191341552320000368084297377590972413770735200004108428908414909144572061556000034365124741040839521574819020000212760336650006960123994184500009477014260566301442915967045000287612140988660262193873203750053303942222088194069911603125045564337691106946230659765625189085664349007369827085937109375
Technical details
Node id
n-bc2e0a7152956095 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d47399ee7119df41

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: 0,               1;
  2: 0,               6,                9;
  3: 0,             120,              330,               225;
  4: 0,            5040,            19740,             25620,             11025;
  5: 0,          362880,          1829520,           3443580,           2869020,            893025;
  6: 0,        39916800,        245155680,         600165720,         732265380,         445384170,         108056025;
  7: 0,      6227020800,      44990225280,      135057162240,      215661405960,      193232779740,       92126664630,       18261468225;
  8: 0,   1307674368000,   10849026988800,    38484804758400,    75676756755600,    89102700086400,    62823975013800,    24563407869000,    4108830350625;
  9: 0, 355687428096000, 3328853233305600, 13603235694048000, 31706186412700800, 46106517125869200, 42838801528723200, 24837375719356200, 8216426729511000, 1187451971330625;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4*n^2-6*n+2+(8*n^2-10*n+3)*x+(4*n^2-4*n+1)*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[((4*n^2 - 6*n + 2 + (8*n^2 - 10*n + 3)*x + (4*n^2 - 4*n + 1)*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] = ((4*n^2 - 6*n + 2 + (8*n^2 - 10*n + 3)*x + (4*n^2 - 4*n + 1)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.