Real-rooted atlas

A038301 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 495\,x + 324\,x^{2} + 297\,n x + 324\,n x^{2} + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 9801\,x + 15147\,x^{2} + 6561\,x^{3} + 3267\,n x + 8019\,n x^{2} + 4374\,n x^{3} + 891\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 792\,x + 729\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01217927291642
1133113068240571166450120
2146411916645292545132161640251412800
316105126353809702990141134409021375212576437760000
417715613478701616009933531049568029770537514030042426040609971200000
5194871714464333722465529759597704184076411046255401566324200512689330611001624262400000
621435888156123052483616110313210519593638416810430175015844594550488225583292269376814083486784401592384000000
7235794769169452277444511421315724173573295033633284651746503921537151224291144424863613334152296963451916556993874204890014197760000000

All 15 rows

Technical details
Node id
n-fa30cc9ac4ed5225 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4d60fe62fd169372

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          121,            792,            729;
  1:         1331,          13068,          24057,           11664;
  2:        14641,         191664,         529254,          513216,           164025;
  3:       161051,        2635380,        9702990,        14113440,          9021375,          2125764;
  4:      1771561,       34787016,      160099335,       310495680,        297705375,        140300424,         26040609;
  5:     19487171,      446433372,     2465529759,      5977041840,       7641104625,       5401566324,       2005126893,        306110016;
  6:    214358881,     5612305248,    36161103132,    105195936384,     168104301750,     158445945504,      88225583292,      26937681408,      3486784401;
  7:   2357947691,    69452277444,   511421315724,   1735732950336,    3328465174650,    3921537151224,    2911444248636,    1333415229696,    345191655699,     38742048900;
  8:  25937424601,   848861168760,  7032043091205,  27275803505280,   61021861535250,   86273817326928,   80064716837490,   48891891755520,  18985541063445,   4261625379000,   421900912521;
  9: 285311670611, 10271220141996, 94541912670645, 412546528017360, 1054806463680750, 1739855316093048, 1937566147467258, 1478979725604480, 765750156225615, 257828335429500, 51050010415041, 4518872583696;
%F Formula
The row polynomials satisfy (121+(297*n+495)*x+(81*n^2+324*n+324)*x^2)*P(n,x) = (1331+(3267*n+9801)*x+(891*n^2+8019*n+15147)*x^2+(729*n^2+4374*n+6561)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 792*x + 729*x^2.
%t Mathematica
P[0, x_] := 121 + 792*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (3267*n + 9801)*x + (891*n^2 + 8019*n + 15147)*x^2 + (729*n^2 + 4374*n + 6561)*x^3)*P[n - 1, x])/(121 + (297*n + 495)*x + (81*n^2 + 324*n + 324)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 792*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (3267*n + 9801)*x + (891*n^2 + 8019*n + 15147)*x^2 + (729*n^2 + 4374*n + 6561)*x^3)*v[i - 1])/(121 + (297*n + 495)*x + (81*n^2 + 324*n + 324)*x^2)); 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.