Real-rooted atlas

A038281 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 6\,x + 3\,n x\right)\,P_{n}(x) = \left(121 + 132\,x + 27\,x^{2} + 33\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 121 + 132\,x + 27\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
012113227280
1133121788911084508
2146413194419602475240571344
31610514392303593701306802227514581114064
4177156157978365929605287496073507596228510317210368
51948717174405562913159175534298018866925370477839293117496263533760
62143588819353842081339300116974036448415072350108673488172889641539648590494005713152
723579476911157537957418941530212160716013928218432530268966882857053581276212576584585119683060507351296

All 15 rows

Technical details
Node id
n-766f173e7ea37394 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3843d29729075732

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:          121,           132,            27;
  1:         1331,          2178,           891,           108;
  2:        14641,         31944,         19602,          4752,           405;
  3:       161051,        439230,        359370,        130680,         22275,          1458;
  4:      1771561,       5797836,       5929605,       2874960,        735075,         96228,         5103;
  5:     19487171,      74405562,      91315917,      55342980,      18866925,       3704778,       392931,       17496;
  6:    214358881,     935384208,    1339300116,     974036448,     415072350,     108673488,     17288964,     1539648,       59049;
  7:   2357947691,   11575379574,   18941530212,   16071601392,    8218432530,    2689668828,    570535812,    76212576,     5845851,     196830;
  8:  25937424601,  141476861460,  260446040415,  252553736160,  150671263050,   59172714216,  15689734830,  2794461120,   321521805,   21651300,   649539;
  9: 285311670611, 1711870023666, 3501552321135, 3819875259420, 2604460404150, 1193316403356, 379691582886, 84532448880, 12968046135, 1309903650, 78594219, 2125764;
%F Formula
The row polynomials satisfy (11+(3*n+6)*x)*P(n,x) = (121+(33*n+132)*x+(9*n+27)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 132*x + 27*x^2.
%t Mathematica
P[0, x_] := 121 + 132*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((121 + (33*n + 132)*x + (9*n + 27)*x^2)*P[n - 1, x])/(11 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 121 + 132*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((121 + (33*n + 132)*x + (9*n + 27)*x^2)*v[i - 1])/(11 + (3*n + 6)*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.