Real-rooted atlas

A038328 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 36\,x + 22\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 616\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01966164841296
1274412936203281064846656
2384162414725691845962882342561679616
35378244225760132809602087008016397920515363260466176
47529536709927682789001605843622406887126404329050881133799042176782336
51054135041159548544546644313614316874880224979462402121234931211111230592249435788878364164096
6147578905618552776704102040271872320697997312629942494720791927707648622228913152279368083456548758735362821109907456
720661046784292206233088183672489369667346579435521587455086694424945722790912261336143523841760018925772869143600655361207269217792101559956668416

All 15 rows

Technical details
Node id
n-89a876c443c480e3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-428cb055b0457fc7

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:           196,            616,             484;
  1:          2744,          12936,           20328,            10648;
  2:         38416,         241472,          569184,           596288,           234256;
  3:        537824,        4225760,        13280960,         20870080,         16397920,           5153632;
  4:       7529536,       70992768,       278900160,        584362240,        688712640,         432905088,         113379904;
  5:     105413504,     1159548544,      5466443136,      14316874880,      22497946240,       21212349312,       11111230592,        2494357888;
  6:    1475789056,    18552776704,    102040271872,     320697997312,     629942494720,      791927707648,      622228913152,      279368083456,       54875873536;
  7:   20661046784,   292206233088,   1836724893696,    6734657943552,   15874550866944,    24945722790912,    26133614352384,    17600189257728,     6914360065536,     1207269217792;
  8:  289254654976,  4545430292480,  32142685639680,  134693158871040,  370406186895360,   698480238145536,   914676502333440,   821342165360640,   484005204587520,   169017690490880,   26559922791424;
  9: 4049565169664, 69999626504192, 549997065390080, 2592843308267520, 8148936111697920, 17927659445735424, 28172036271869952, 31621673366384640, 24845600502159360, 13014362167797760, 4090228109879296, 584318301411328;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (14+36*x+22*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 616*x + 484*x^2.
%t Mathematica
P[0, x_] := 196 + 616*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14 + 36*x + 22*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] = 196 + 616*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14 + 36*x + 22*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

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. [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.