Real-rooted atlas

A109692 → basis A007318 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4 + 12\,x + 9\,x^{2} - 16\,n - 44\,n x - 30\,n x^{2} + 20\,n^{2} + 48\,n^{2} x + 28\,n^{2} x^{2} - 8\,n^{3} - 16\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2 + 6\,x - 12\,n - 16\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
210616
310013645281
41136248817884205832
5148164532851468256564725141993
622003286988813655281062736409470623703990024
73679488178992643608020838547456230020407260912945945127415313
86854860839677286498001676813383634561090797440530211312142201080162162004563127728
91409347584947256422427758009856463042556164807457736031796071392130754771283054655440310134825181255093425
10317170483202430064926728251282775041629288028672206121761808017320217916489663224584163450177550087149595653065472907507911762717600
11775808778240667825886822425807526624256589479284008968811559835040090047163391872636936279805283078212961216097239555057241812279382200151242416325376535519310825
122049965162496019596541663641684978056993587222061810980986883809527301935360459312243587596839449126225479042413003617608576102986253585201629199579031908049480486361460379480971870019408126336515000
13582040706088960612008341379481629442997942198272856867275484282881679908691659033602337070023908359682365337992326279041754498832734773769464215267739361636199183210852080931637687917266014481328625442001027760965481251077222025399865625
141767445713911808020279253371505868810722673206402416643459176297275129856759567139140474880011986700227918675968139847169538249143041221135396512131328079792704793988965443852841710225666720133597429397914125631495736641050504045235616932765350298841265347625064061621026639651800
Technical details
Node id
n-658eb7080e7f6be8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ca11cc4fe63ef8ce

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:          0;
  1:          1;
  2:         10,          6;
  3:        100,        136,          45;
  4:       1136,       2488,        1788,         420;
  5:      14816,      45328,       51468,       25656,        4725;
  6:     220032,     869888,     1365528,     1062736,      409470,       62370;
  7:    3679488,   17899264,    36080208,    38547456,    23002040,     7260912,      945945;
  8:   68548608,  396772864,   980016768,  1338363456,  1090797440,   530211312,   142201080,   16216200;
  9: 1409347584, 9472564224, 27758009856, 46304255616, 48074577360, 31796071392, 13075477128, 3054655440, 310134825;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(16*n^3-48*n^2+44*n-12)*x+(8*n^3-28*n^2+30*n-9)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(8*n^2-16*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1])/(2*n - 3)); 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.