Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(9 + 6\,x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
196116
28110854121256
3729145812155401351814096
465611749620412136085670151225224165536
559049196830295245262440153090612361701032404053011048576
65314412125764389723443302603247695173210467359619245640095594059436116777216
7478296922320522483611316448150859108049394053661970268375057842189187486486810819828819421268435456
8430467212295825125739562808928208809672226207737780964728643922251735208444007025019280583783210614241474201512010804814294967296
9387420489232452293465861483131170870811214635885140136601594649865670724563752612825838661429569874602870962386959908813533156208202424786022032137754168719476736
103486784401232452293407360989291014721978582020856136324522246545412818538787844012359191896066945622770297536101201090965704433059566808264891701695362402825604037674723924453078017106011099511627776
1131381059609230127770466805447196631178988265918028339808770353400777052442321184499397324471199954081529449997130793048146660343654196886124965162504381837996549790717860209801097037297972854392877639916259251541580207966117592186044416
1228242953648122594362918488661172452084211717548828723705057104502649400761393368521452481374444469592697495231659614940591187612532981289380626649064397965979051214370993687964421844211681158102055442573560123248254252317569294489812048410328472860706546482484721281474976710656
1325418658283292202950384551891789599356325244772264950200469146841154550688082033693340802762372642230764535592992600605257344452475403504896301650228652774570935110861951307120461924797113001658191579380051324977457001368666065520313652640015615005176501025008627514386086001678376701598454012109507020029257814503599627370496
142287679245496121351672957963696082528310836227757174845352685782744759448475925239161511756011822500397094660123854766064801201083729203067010580276237264223005084161693400790277317910549134013095567998153555372540717191201918764541854005969489685768016167367898955380408656446077490652239013594851270020392276905258949548027464346023882040165847588452340284172057594037927936
Technical details
Node id
n-dadebac29001ef09 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-12929af9fbcc0757

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:         9,          6,          1;
  2:        81,        108,         54,          12,           1;
  3:       729,       1458,       1215,         540,         135,          18,          1;
  4:      6561,      17496,      20412,       13608,        5670,        1512,        252,         24,          1;
  5:     59049,     196830,     295245,      262440,      153090,       61236,      17010,       3240,        405,        30,         1;
  6:    531441,    2125764,    3897234,     4330260,     3247695,     1732104,     673596,     192456,      40095,      5940,       594,       36,        1;
  7:   4782969,   22320522,   48361131,    64481508,    59108049,    39405366,   19702683,    7505784,    2189187,    486486,     81081,     9828,      819,      42,      1;
  8:  43046721,  229582512,  573956280,   892820880,   967222620,   773778096,  472864392,  225173520,   84440070,  25019280,   5837832,  1061424,   147420,   15120,   1080,    48,    1;
  9: 387420489, 2324522934, 6586148313, 11708708112, 14635885140, 13660159464, 9865670724, 5637526128, 2583866142, 956987460, 287096238, 69599088, 13533156, 2082024, 247860, 22032, 1377, 54, 1;
%F Formula
The row polynomials satisfy P(n,x) = (9+6*x+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[(9 + 6*x + 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] = (9 + 6*x + 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.