Real-rooted atlas

A038305 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(13 + 29\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 416\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0169416256841
1219781129984409624389
22856114060825958421299265536707281
33712932284880562432069222404259840104857620511149
44826809356441281096742401799782401661337608178892816777216594823321
5627485175406026081996071168409450496050393907203721396224152672665626843545617249876309
68157307218031810176345985669128516570316813102415872012900840243279389786112279172874244294967296500246412961
710604499373117465223824578290332672166073121177630659653140483773495771136309620165836816331613143045025111736326871947673614507145975869
813785849184916967198996809397217905920308421510758406642924847104098110890049536100626553896960707703236198403266322628608089335319756801099511627776420707233300201
9179216039403724263094565424149311351171840551303450480640135705464733696023383095461806082877919441453056253003906940928015569471196364806387475362611201572301627719681759218604441612200509765705829
102329808512248134409479565510423292570782807049555926474997760264625656230707205211089845773926474825905477779456789372189655695366072093766582272033214871885578240122639526962135042744381022928896281474976710656353814783205469041
113028751065922534846001705476048357858587481308161614951574274621444969081766998835201100842729919741952180651114653496115222233983341968752642052367693104807936140332833716568064069086933522002739223190019643749171247569271064100864450359962737049610260628712958602189
123937376385699289678440238766646725427521910133173762672010786527100928904372881593788006422261486316154781696410981285836703662085780835668911875686462255153357512507392510811514728307752963143455475251124633614068611917207830528432880366683317862481965513218143027272057594037927936297558232675799463481
13511858930140907579449703325678293608141282865199760640434201752810653900801603206471916260556804340989831650182430728904594526461912678401409078694297269698560173425070067356270592016601374228670001971201225947635347938607104685844830963881738240281372238344156610560799163753876894515201405123083739594752011529215046068469768629188747598184440949
146654166091831798411310358861160723379212095620256868215808069472280449704624128027788912179881849651208208417136211254050816185215566150407783710723256537426820356636672045090518217512630353920493297977080480058572804249951802539520504627228531144968097480310784146313563938961437491205540868693546468638720146132800708917854208023980767295822417100818446744073709551616250246473680347348787521
Technical details
Node id
n-4070f46c6317d3ae · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b7ed2a977b2f213e

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           169,            416,             256;
  1:          2197,           8112,            9984,            4096;
  2:         28561,         140608,          259584,          212992,            65536;
  3:        371293,        2284880,         5624320,         6922240,          4259840,          1048576;
  4:       4826809,       35644128,       109674240,       179978240,        166133760,         81788928,         16777216;
  5:      62748517,      540602608,      1996071168,      4094504960,       5039390720,       3721396224,       1526726656,        268435456;
  6:     815730721,     8031810176,     34598566912,     85165703168,     131024158720,     129008402432,      79389786112,      27917287424,       4294967296;
  7:   10604499373,   117465223824,    578290332672,   1660731211776,    3065965314048,    3773495771136,    3096201658368,    1633161314304,     502511173632,     68719476736;
  8:  137858491849,  1696719899680,   9397217905920,  30842151075840,   66429248471040,   98110890049536,  100626553896960,   70770323619840,   32663226286080,   8933531975680,   1099511627776;
  9: 1792160394037, 24263094565424, 149311351171840, 551303450480640, 1357054647336960, 2338309546180608, 2877919441453056, 2530039069409280, 1556947119636480, 638747536261120, 157230162771968, 17592186044416;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (13+29*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 416*x + 256*x^2.
%t Mathematica
P[0, x_] := 169 + 416*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((13 + 29*x + 16*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] = 169 + 416*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((13 + 29*x + 16*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. [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.