Real-rooted atlas

A038333 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(33 + 42\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1089 + 2772\,x + 1764\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01089277217645625
13593713721417463674088421875
211859216037416115259769779616311169631640625
3391353932490434106339286808068183205134298401306912322373046875
412914679699862119036313794696605325000912050829554160258768639365489031744177978515625
5426184429773796915828861449731498292307518802668039138756703202988777784608126796633286423053933324813348388671875
6140640861824114319796840272637881859248481623699278087042583157942411202630124450455041673715559380486086238397747296826519964161001129150390625
746411484401953531622457695098270644160281140880373114265308481534395817792252819528674044628672165697840378667529038064020654592287574764293555240667138384947275084686279296875

All 15 rows

Technical details
Node id
n-cad214743f90c94e · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-64aa42accc9ceb8f

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:              1089,               2772,                1764;
  1:             35937,             137214,              174636,                74088;
  2:           1185921,            6037416,            11525976,              9779616,              3111696;
  3:          39135393,          249043410,           633928680,            806818320,            513429840,            130691232;
  4:        1291467969,         9862119036,         31379469660,          53250009120,          50829554160,          25876863936,           5489031744;
  5:       42618442977,       379691582886,       1449731498292,        3075188026680,        3913875670320,        2988777784608,        1267966332864,         230539333248;
  6:     1406408618241,     14319796840272,      63788185924848,      162369927808704,      258315794241120,      263012445045504,      167371555938048,       60862383977472,        9682651996416;
  7:    46411484401953,    531622457695098,    2706441602811408,     8037311426530848,    15343958177922528,    19528674044628672,    16569784037866752,     9038064020654592,     2875747642935552,      406671383849472;
  8:  1531578985264449,  19492823448820260,  111640716115970580,   378901824393597120,   843917699785739040,  1288892486945492352,  1367007183124007040,   994187042272005120,   474498361084366080,   134201556670325760,   17080198121677824;
  9: 50542106513726817, 707589491192175438, 4502842216677480060, 17192670281859469320, 43763160717460467360, 77977995460202287296, 99244721494802911104, 90222474086184464640, 57414301691208295680, 24357582535664125440, 6200111918169050112, 717368321110468608;
%F Formula
The row polynomials satisfy P(n,x) = (33+42*x)*P(n-1,x) for n >= 1, with P(0,x) = 1089 + 2772*x + 1764*x^2.
%t Mathematica
P[0, x_] := 1089 + 2772*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[(33 + 42*x)*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] = 1089 + 2772*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = (33 + 42*x)*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. [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.