Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 8\,x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1168125
225625696161625
3409661443840128024024115625
46553613107211468857344179203584448321390625
510485762621440294912019660808601602580485376076807204019765625
61677721650331648692060165767168032440320129761283784704811008126720140801056481244140625
7268435456939524096152672665615267266561049624576524812288196804608562298881230028820500482562562329614565616103515625
84294967296171798691843221225472037580963840305345331201832071987283969966082998927360843448320187432960328007684472832465920358401920641152587890625
968719476736309237645312657129996288876173328384821412495360574988746752311452237824133479530496458835886081274544128028677242885214044167603814487736327833605222424487213814697265625
1010995116277765497558138880130567005798401958505086976020809116549120166472932392961040455827456052022791372802113425899520704475299840193730707456440297062408255569920127008768015876096015876096124032072960304080195367431640625
11175921860444169675702324428825398718601625642331197669376050268297232384045241467509145632046039485644818312022563225685837605765120333812911308801084891961753629587962593286780574760961303956684802095644672027941928963056148482696601618726409856036968812384185791015625
122814749767106561688849860263936485544334825881689016461384744961168341055674777611683410556747776924936669075865659460214440591363158823892156416140392172984729652647064869273616751338822041645368209309696104695867637762056525971456342754328576481998274565670567936551305216435240962720256129536441696159604644775390625
134503599627370496292733975779082249147936743096320018295873486192640026300318136401920028930349950042112025314056206286848018081468718776320010735872051773440053679360258867200228137281100185608295901130915840259246910341120069797245091840016202931896320032405863792640556975783936081908203520010238525440010777395200943022080673587203827200166400520010411490116119384765625
147205759403792793650440315826549555217023606591460474883688448094816436224576320014815068160069158401777808179206627680170373283840520746299100758016034173975878487244801898554215471513600901813252348968960368923603233669120130660442811924480402032131728998401076871781416960025127008233062405103923547340809006923907072013760578191360181060239360020369276928019399311360154312704010063872052416002096646048112137252902984619140625
Technical details
Node id
n-a40ac0f77c5e83c7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-268f03d6dbc16036

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:          16,            8,            1;
  2:         256,          256,           96,           16,            1;
  3:        4096,         6144,         3840,         1280,          240,           24,            1;
  4:       65536,       131072,       114688,        57344,        17920,         3584,          448,           32,           1;
  5:     1048576,      2621440,      2949120,      1966080,       860160,       258048,        53760,         7680,         720,          40,          1;
  6:    16777216,     50331648,     69206016,     57671680,     32440320,     12976128,      3784704,       811008,      126720,       14080,       1056,        48,        1;
  7:   268435456,    939524096,   1526726656,   1526726656,   1049624576,    524812288,    196804608,     56229888,    12300288,     2050048,     256256,     23296,     1456,      56,      1;
  8:  4294967296,  17179869184,  32212254720,  37580963840,  30534533120,  18320719872,   8396996608,   2998927360,   843448320,   187432960,   32800768,   4472832,   465920,   35840,   1920,    64,    1;
  9: 68719476736, 309237645312, 657129996288, 876173328384, 821412495360, 574988746752, 311452237824, 133479530496, 45883588608, 12745441280, 2867724288, 521404416, 76038144, 8773632, 783360, 52224, 2448, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (16+8*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[(16 + 8*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] = (16 + 8*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.