Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 48\,x + 36\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1164836100
2256153634563456129610000
3409636864138240276480311040186624466561000000
46553678643241287681238630423224320278691842090188889579521679616100000000
5104857615728640106168320424673280111476736020065812482508226560214990848012093235204031078406046617610000000000
61677721630198988824914165761245708288042042654720100902371328176579149824227030335488212840939520141893959680638522818561741425868821767823361000000000000
7268435456563714457654962159616329772957696136031345049640809403514889182115790848157407699271682065976052940820659760529408154948203970568451720216576316939508121673139886489678364164096100000000000000

All 15 rows

Technical details
Node id
n-c1a4b2af6b8ff4f9 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-6b852c5cd7d4c499

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,            48,             36;
  2:         256,          1536,           3456,            3456,             1296;
  3:        4096,         36864,         138240,          276480,           311040,           186624,             46656;
  4:       65536,        786432,        4128768,        12386304,         23224320,         27869184,          20901888,           8957952,           1679616;
  5:     1048576,      15728640,      106168320,       424673280,       1114767360,       2006581248,        2508226560,        2149908480,        1209323520,          403107840,           60466176;
  6:    16777216,     301989888,     2491416576,     12457082880,      42042654720,     100902371328,      176579149824,      227030335488,      212840939520,       141893959680,        63852281856,        17414258688,         2176782336;
  7:   268435456,    5637144576,    54962159616,    329772957696,    1360313450496,    4080940351488,     9182115790848,    15740769927168,    20659760529408,     20659760529408,     15494820397056,      8451720216576,      3169395081216,       731398864896,       78364164096;
  8:  4294967296,  103079215104,  1159641169920,   8117488189440,   39572754923520,  142461917724672,   391770273742848,   839507729448960,  1416669293445120,   1888892391260160,   1983337010823168,   1622730281582592,   1014206425989120,    468095273533440,   150459195064320,    30091839012864,    2821109907456;
  9: 68719476736, 1855425871872, 23656679866368, 189253438930944, 1064550593986560, 4471112494743552, 14531115607916544, 37365725848928256, 77066809563414528, 128444682605690880, 173400321517682688, 189163987110199296, 165518488721424384, 114589722960986112, 61387351586242560, 24554940634497024, 6906077053452288, 1218719480020992, 101559956668416;
%F Formula
The row polynomials satisfy P(n,x) = (16+48*x+36*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 + 48*x + 36*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 + 48*x + 36*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.