Real-rooted atlas

A034870 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1364816100
2129634563456153625610000
3466561866243110402764801382403686440961000000
41679616895795220901888278691842322432012386304412876878643265536100000000
5604661764031078401209323520214990848025082265602006581248111476736042467328010616832015728640104857610000000000
62176782336174142586886385228185614189395968021284093952022703033548817657914982410090237132842042654720124570828802491416576301989888167772161000000000000
7783641640967313988648963169395081216845172021657615494820397056206597605294082065976052940815740769927168918211579084840809403514881360313450496329772957696549621596165637144576268435456100000000000000

All 15 rows

Technical details
Node id
n-fd18f6eb36bb6ac9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
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:              36,               48,               16;
  2:            1296,             3456,             3456,              1536,               256;
  3:           46656,           186624,           311040,            276480,            138240,              36864,               4096;
  4:         1679616,          8957952,         20901888,          27869184,          23224320,           12386304,            4128768,             786432,              65536;
  5:        60466176,        403107840,       1209323520,        2149908480,        2508226560,         2006581248,         1114767360,          424673280,          106168320,           15728640,           1048576;
  6:      2176782336,      17414258688,      63852281856,      141893959680,      212840939520,       227030335488,       176579149824,       100902371328,        42042654720,        12457082880,        2491416576,         301989888,          16777216;
  7:     78364164096,     731398864896,    3169395081216,     8451720216576,    15494820397056,     20659760529408,     20659760529408,     15740769927168,      9182115790848,      4080940351488,     1360313450496,      329772957696,       54962159616,       5637144576,        268435456;
  8:   2821109907456,   30091839012864,  150459195064320,   468095273533440,  1014206425989120,   1622730281582592,   1983337010823168,   1888892391260160,   1416669293445120,    839507729448960,   391770273742848,   142461917724672,    39572754923520,    8117488189440,    1159641169920,    103079215104,     4294967296;
  9: 101559956668416, 1218719480020992, 6906077053452288, 24554940634497024, 61387351586242560, 114589722960986112, 165518488721424384, 189163987110199296, 173400321517682688, 128444682605690880, 77066809563414528, 37365725848928256, 14531115607916544, 4471112494743552, 1064550593986560, 189253438930944, 23656679866368, 1855425871872, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (36+48*x+16*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 + 48*x + 16*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 + 48*x + 16*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.