Real-rooted atlas

A133214 → basis A013609 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-25 - 80\,x - 64\,x^{2} - 25\,n - 80\,n x - 64\,n x^{2}\right)\,P_{n-2}(x) + \left(21 + 24\,x + 14\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 61 + 128\,x + 64\,x^{2}\)
\(P_{1}(x) = 595 + 1800\,x + 1728\,x^{2} + 512\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
06112864253
1595180017285124635
26145243203456020480409689601
365527320840605440542720225280327681792575
471290941621769806400118169607557120235929626214436677005
578636675333165615075244822856960019855360097026048238551042097152762049275
687615745676981760223281049640832368644492902400300941312011744051202348810241677721616009023745
798372669585308325203216278016068896235520918661201927840530432042278584320135895449602264924160134217728339112270575
81111221078110687229504045340604064011135648563201745348935680181276036300812521727590405601912422401517499187202147483648010737418247229727199773
9126142119187133248993600862827534000001740474844416031361590149120384096892354563241135728230418664862515200708392779776016461804339202007897210888589934592154933120968795
101437751935361165477002503688584534736870426481044817920053938198702080076134389435596875977575523942453638205748019226321816125440086221468467200174332722544641855425871872687194767363334321989154689
11164433809947752048118858203601159352487191808394112906438451289547497847808001431651529211904016503625073295360138160775406551048338786228371456354641153884160010164684351078401809040225075201697371075379254975581388872012798200637855
12188609259215725252771086576960015503673689358400575845204908646401444004461665525762579522156193382403374329898690150403270066140558131202346276196712448001231007763849543684598598398377984011662725994250240184477435297792015393162788864043980465111041559976990077534253
1321688330848413953111879380920020020558524851940320082834190317948160022723928346921984004486256768565411840656730595047374848072346340919921868806021007410871664640376421168527179776017396825022403706885771290578046156801307288401661132801852951970709504013853846509977603518437208883233880514009401215675
1424994269200292865382276635994664960270632901450700800011756883036269772800350251824045899776007573608661587591168012268703822456553472015146414241071759360014352618115345416192010427509308479504384057551713149581787136236840511088298557447041409903230976000143587422494523392018366242230370304012384898975268864281474976710656737493584032813497345
Technical details
Node id
n-219b4be2425389c0 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-2c6cd6b0f6303dd6

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           61,           128,            64;
  1:          595,          1800,          1728,            512;
  2:         6145,         24320,         34560,          20480,           4096;
  3:        65527,        320840,        605440,         542720,         225280,          32768;
  4:       712909,       4162176,       9806400,       11816960,        7557120,        2359296,         262144;
  5:      7863667,      53331656,     150752448,      228569600,      198553600,       97026048,       23855104,        2097152;
  6:     87615745,     676981760,    2232810496,     4083236864,     4492902400,     3009413120,     1174405120,      234881024,      16777216;
  7:    983726695,    8530832520,   32162780160,    68896235520,    91866120192,    78405304320,    42278584320,    13589544960,    2264924160,     134217728;
  8:  11112210781,  106872295040,  453406040640,  1113564856320,  1745348935680,  1812760363008,  1252172759040,   560191242240,  151749918720,   21474836480,   1073741824;
  9: 126142119187, 1332489936008, 6282753400000, 17404748444160, 31361590149120, 38409689235456, 32411357282304, 18664862515200, 7083927797760, 1646180433920, 200789721088, 8589934592;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(25*n+25+(80*n+80)*x+(64*n+64)*x^2)*P(n-2,x) + (14*n+21+(16*n+24)*x)*P(n-1,x) for n >= 2, with P(0,x) = 61 + 128*x + 64*x^2, P(1,x) = 595 + 1800*x + 1728*x^2 + 512*x^3.
%t Mathematica
P[0, x_] := 61 + 128*x + 64*x^2;
P[1, x_] := 595 + 1800*x + 1728*x^2 + 512*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(25*n + 25 + (80*n + 80)*x + (64*n + 64)*x^2)*P[n - 2, x] + (14*n + 21 + (16*n + 24)*x)*P[n - 1, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 61 + 128*x + 64*x^2; v[2] = 595 + 1800*x + 1728*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(25*n + 25 + (80*n + 80)*x + (64*n + 64)*x^2)*v[i - 2] + (14*n + 21 + (16*n + 24)*x)*v[i - 1])/(n + 2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023