Real-rooted atlas

A038321 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 64\,x + 46\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 1656\,x + 2116\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0324165621164096
158324471211426497336262144
2104976107308841135047008192447745616777216
31889568241444801234051203153686404029710402059629761073741824
4340122245215207683331938240113532710402176043616022244001408947429689668719476736
56122200321095193612883964843648357628037760913938318720140137208870411937614088964358176572164398046511104
6110199605762252969717762015156247552102996874874883290177947392067265860257792859508214405126275774263910420047612231936281474976710656
71983592903684562263678464466364731576322780915621621761066017654955008272426734044057646413443577876485083377153767424324771318157363292219016266905618014398509481984

All 15 rows

Technical details
Node id
n-952f9bc3bca084fe · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-aa5fb0cc387fb3c0

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            324,             1656,              2116;
  1:           5832,            44712,            114264,              97336;
  2:         104976,          1073088,           4113504,            7008192,            4477456;
  3:        1889568,         24144480,         123405120,          315368640,          402971040,           205962976;
  4:       34012224,        521520768,        3331938240,        11353271040,        21760436160,         22244001408,          9474296896;
  5:      612220032,      10951936128,       83964843648,       357628037760,       913938318720,       1401372088704,       1193761408896,         435817657216;
  6:    11019960576,     225296971776,     2015156247552,     10299687487488,     32901779473920,      67265860257792,      85950821440512,       62757742639104,       20047612231936;
  7:   198359290368,    4562263678464,    46636473157632,    278091562162176,   1066017654955008,    2724267340440576,    4641344357787648,     5083377153767424,     3247713181573632,      922190162669056;
  8:  3570467226624,   91245273569280,  1049320646046720,   7150925884170240,  31980529648650240,   98073624255860736,  208860496100444160,   305002629226045440,   292294186341626880,   165994229280430080,   42420747482776576;
  9: 64268410079232, 1806656416671744, 23085054213027840, 176985415633213440, 904592124347535360, 3236429600443404288, 8270875645577588736, 15097630146689249280, 19291416298547374080, 16433428698762577920, 8399308001589762048, 1951354384207722496;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (18+64*x+46*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1656*x + 2116*x^2.
%t Mathematica
P[0, x_] := 324 + 1656*x + 2116*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + 64*x + 46*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1656*x + 2116*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + 64*x + 46*x^2)*v[i - 1])/(1 + x)); 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

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.