Real-rooted atlas

A038327 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116925
225628881625
340966912388872915625
465536147456124416466566561390625
51048576294912033177601866240524880590499765625
616777216566231047962624059719680251942405668704531441244140625
72684354561056964608178362777616721510409405849603174474245952139247829696103515625
842949672961932735283238050725888428070666243009871872013544423424380936908861222003243046721152587890625
96871947673634789235097678275778969610273695989768668430991364875992432641828497162244407984230461987278243874204893814697265625
1010995116277766184752906240156551557939202348273369088023115815976960156031757844487313988648960235092492288049589822592061987278240348678440195367431640625
1117592186044416108851651149824306145268858880516620141199360581197658849280457693156343808257452400443392103440696606720290926959206405454880485120613674054576313810596092384185791015625
122814749767106561899956092796928587798916209049611021229678919680139487438123827201255386943114444882384768141885443972122749698048139644940419072034911235104768058912709239296602516344492828242953648159604644775390625
13450359962737049632932572275146752111147431428620288229241577321529344322370968108400640326400605209755648244800453907316736137700255322865664580922952143339521815384225447936040846145072578566266169982725125874534358804825418658283291490116119384765625
147205759403792793656745355304868249620747520533342453764668192120002052096722110968562817433681237483963316961286854412709404868608440640817033170124821687790213351342088132921330006753282287384124064399364678740253768089665794784818613765693779455456962287679245496137252902984619140625
Technical details
Node id
n-b2dfc4991f197cd3 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f3c17ca3ebbc301e

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:          16,            9;
  2:         256,          288,           81;
  3:        4096,         6912,         3888,           729;
  4:       65536,       147456,       124416,         46656,         6561;
  5:     1048576,      2949120,      3317760,       1866240,       524880,        59049;
  6:    16777216,     56623104,     79626240,      59719680,     25194240,      5668704,       531441;
  7:   268435456,   1056964608,   1783627776,    1672151040,    940584960,    317447424,     59521392,     4782969;
  8:  4294967296,  19327352832,  38050725888,   42807066624,  30098718720,  13544423424,   3809369088,   612220032,   43046721;
  9: 68719476736, 347892350976, 782757789696, 1027369598976, 866843099136, 487599243264, 182849716224, 44079842304, 6198727824, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (16+9*x)*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[(16 + 9*x)*P[n - 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] = (16 + 9*x)*v[i - 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

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.