Real-rooted atlas

A059364 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 8\,n x - 16\,n x^{2} + 4\,n^{2} + 24\,n^{2} x + 32\,n^{2} x^{2} - 4\,n^{3} - 16\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 8\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
24464108
3488150411523144
46072291524646424576106264
585692563552138566415093766144004158684
613634761143939238301504639631365326848017694720186030708
72423592024794553610549770242389369856303773696020552417285780275209387534544
8476875440576617299229835995136856316405761472163225601515882086408655549235221139292160528209999856
9103011726601438923185288782572491523059128937472665053395456092399210987528011259117568396278680780885614133248032812221988980
10242513788860384583250188827076941343872111082094308352292617659919360513263972892672599432759410688449447511195648196304374333440380507258880002231364385582780
11618278680908010979536096310487659459603366441434020612659201283972136510464027255283820273664401345040749363204048019723072307226762652413853696104722694406144001841655132979200164922258283556760
12169748909101800333843155774928029820130890185088159687105677802496569597157803571200142095004613686886425296584958087331843213669154307440640285499696080538828816891481980875571205989772047889203209643575969054720013166448394463865480
1349943249319411001077915386058177601065541195539441216637908092835350016025758785129753584640739085138926383022081545050821858058403842371030577743627550722650845634687307612162105623813060086988801127916614599091159043658210876512534528054325477959008256001129286109765665929420
141567890736401969003685319011946118240399551839135324871042645537561891964487681193540433747598965760387433726360595165184093093369577074051153921676431393534809368166422624944980378045644800226004596737063885209601624160087078750021222479514262906485373665282376858484498728222720327624420922019020800103572794359538583653700
Technical details
Node id
n-aef0d445e9ade36a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f78b9b66a62f04a5

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:           4;
  2:          44,           64;
  3:         488,         1504,         1152;
  4:        6072,        29152,        46464,         24576;
  5:       85692,       563552,      1385664,       1509376,        614400;
  6:     1363476,     11439392,     38301504,      63963136,      53268480,      17694720;
  7:    24235920,    247945536,   1054977024,    2389369856,    3037736960,    2055241728,     578027520;
  8:   476875440,   5766172992,  29835995136,   85631640576,  147216322560,  151588208640,   86555492352,   21139292160;
  9: 10301172660, 143892318528, 878257249152, 3059128937472, 6650533954560, 9239921098752, 8011259117568, 3962786807808, 856141332480;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(16*n^3-24*n^2+8*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 1)); 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.