Real-rooted atlas

A038327 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1131629
2169416256841
3219781129984409624389
42856114060825958421299265536707281
53712932284880562432069222404259840104857620511149
64826809356441281096742401799782401661337608178892816777216594823321
7627485175406026081996071168409450496050393907203721396224152672665626843545617249876309
88157307218031810176345985669128516570316813102415872012900840243279389786112279172874244294967296500246412961
910604499373117465223824578290332672166073121177630659653140483773495771136309620165836816331613143045025111736326871947673614507145975869
1013785849184916967198996809397217905920308421510758406642924847104098110890049536100626553896960707703236198403266322628608089335319756801099511627776420707233300201
11179216039403724263094565424149311351171840551303450480640135705464733696023383095461806082877919441453056253003906940928015569471196364806387475362611201572301627719681759218604441612200509765705829
122329808512248134409479565510423292570782807049555926474997760264625656230707205211089845773926474825905477779456789372189655695366072093766582272033214871885578240122639526962135042744381022928896281474976710656353814783205469041
133028751065922534846001705476048357858587481308161614951574274621444969081766998835201100842729919741952180651114653496115222233983341968752642052367693104807936140332833716568064069086933522002739223190019643749171247569271064100864450359962737049610260628712958602189
143937376385699289678440238766646725427521910133173762672010786527100928904372881593788006422261486316154781696410981285836703662085780835668911875686462255153357512507392510811514728307752963143455475251124633614068611917207830528432880366683317862481965513218143027272057594037927936297558232675799463481
Technical details
Node id
n-576df5f1b88caa47 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b7ed2a977b2f213e

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:          13,           16;
  2:         169,          416,          256;
  3:        2197,         8112,         9984,          4096;
  4:       28561,       140608,       259584,        212992,         65536;
  5:      371293,      2284880,      5624320,       6922240,       4259840,       1048576;
  6:     4826809,     35644128,    109674240,     179978240,     166133760,      81788928,      16777216;
  7:    62748517,    540602608,   1996071168,    4094504960,    5039390720,    3721396224,    1526726656,     268435456;
  8:   815730721,   8031810176,  34598566912,   85165703168,  131024158720,  129008402432,   79389786112,   27917287424,   4294967296;
  9: 10604499373, 117465223824, 578290332672, 1660731211776, 3065965314048, 3773495771136, 3096201658368, 1633161314304, 502511173632, 68719476736;
%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) = 1.
%t Mathematica
P[0, x_] := 1;
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] = 1; 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

Candidate equalities ?

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.