Real-rooted atlas

A034870 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(36 + 96\,x + 64\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369664196
2129669121382412288409638416
34665637324812441602211840221184011796482621447529536
416796161791590483607552222953472371589120396361728264241152100663296167772161475789056
560466176806215680483729408017199267840401316249606421059993671345111040543581798402717908992080530636801073741824289254654976
6217678233634828517376255409127424113515167744034054550323207264970735616113010655887361291550352998410762919608320637802643456025512105738246184752906246871947673656693912375296
778364164096146279772979212677580324864676137617326082479171263528966611123369410561322224673882112201481855067750423506216424570882089441459961856139296097330790467537501736140822512500578713646179488366592439804651110411112006825558016

All 15 rows

Technical details
Node id
n-c30e479d0a74aad5 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-7cfe8b61bc182db1

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:              36,               96,                64;
  2:            1296,             6912,             13824,              12288,               4096;
  3:           46656,           373248,           1244160,            2211840,            2211840,             1179648,               262144;
  4:         1679616,         17915904,          83607552,          222953472,          371589120,           396361728,            264241152,            100663296,             16777216;
  5:        60466176,        806215680,        4837294080,        17199267840,        40131624960,         64210599936,          71345111040,          54358179840,          27179089920,           8053063680,           1073741824;
  6:      2176782336,      34828517376,      255409127424,      1135151677440,      3405455032320,       7264970735616,       11301065588736,       12915503529984,       10762919608320,        6378026434560,        2551210573824,         618475290624,          68719476736;
  7:     78364164096,    1462797729792,    12677580324864,     67613761732608,    247917126352896,     661112336941056,     1322224673882112,     2014818550677504,     2350621642457088,     2089441459961856,     1392960973307904,      675375017361408,      225125005787136,       46179488366592,        4398046511104;
  8:   2821109907456,   60183678025728,   601836780257280,   3744762188267520,  16227302815825920,   51927369010642944,   126933568692682752,   241778226081300480,   362667339121950720,   429827957477867520,   401172760312676352,   291762007500128256,   162090004166737920,    66498463247892480,    18999560927969280,    3377699720527872,     281474976710656;
  9: 101559956668416, 2437438960041984, 27624308213809152, 196439525075976192, 982197625379880960, 3666871134751555584, 10593183278171160576, 24212990350105509888, 44390482308526768128, 65763677494113730560, 78916412992936476672, 76525006538605068288, 59519449530026164224, 36627353556939177984, 17441596931875799040, 6201456686889172992, 1550364171722293248, 243194379878006784, 18014398509481984;
%F Formula
The row polynomials satisfy P(n,x) = (36+96*x+64*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[(36 + 96*x + 64*x^2)*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] = (36 + 96*x + 64*x^2)*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.