Real-rooted atlas

A038301 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(42 + 66\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1764 + 5544\,x + 4356\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
017645544435611664
1740883492725488562874961259712
2311169619559232461039044829932818974736136048896
31306912321026859680322727328050714294403984694560125233257614693280768
4548903174451753727872203318216640426000072960502071514560315587809152826539500161586874322944
5230539333248253593266572811955111138432313110053625604920300842688046391407945344243002613047045455160701056171382426877952
696826519964161217247679549446694862237521922104099560364032413305270785792051958376898785284082443899190272183293399555481636004060626969618509302102818816
7406671383849472575149528587110436152256082618368132558272302934016312458784714058752491006661693520896514387931297974272346424525159860224136095349169945088237626800137999361999004627104432128

All 15 rows

Technical details
Node id
n-bb612dd1f8b33da4 · revision 0 · JSON
Definition source
derived:basis_map_A013610
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:               1764,                 5544,                 4356;
  1:              74088,               349272,               548856,                287496;
  2:            3111696,             19559232,             46103904,              48299328,               18974736;
  3:          130691232,           1026859680,           3227273280,            5071429440,             3984694560,             1252332576;
  4:         5489031744,          51753727872,         203318216640,          426000072960,           502071514560,           315587809152,            82653950016;
  5:       230539333248,        2535932665728,       11955111138432,        31311005362560,         49203008426880,         46391407945344,         24300261304704,          5455160701056;
  6:      9682651996416,      121724767954944,      669486223752192,      2104099560364032,       4133052707857920,       5195837689878528,       4082443899190272,       1832933995554816,        360040606269696;
  7:    406671383849472,     5751495285871104,    36152256082618368,    132558272302934016,     312458784714058752,     491006661693520896,     514387931297974272,     346424525159860224,     136095349169945088,      23762680013799936;
  8:  17080198121677824,   268403113340651520,  1897993444337464320,   7953496338176040960,   21872114929984112640,   41244559582255755264,   54010732786287298560,   48499433522380431360,   28580023325688468480,    9980325605795973120,   1568336880910795776;
  9: 717368321110468608, 12400223836338100224, 97430330142656501760, 459314413529666365440, 1443559585378951434240, 3175831087833693155328, 4990591709452946386944, 5601684571834939822080, 4401323592156024145920, 2305455214938869790720, 724571638980787648512, 103510234140112521216;
%F Formula
The row polynomials satisfy P(n,x) = (42+66*x)*P(n-1,x) for n >= 1, with P(0,x) = 1764 + 5544*x + 4356*x^2.
%t Mathematica
P[0, x_] := 1764 + 5544*x + 4356*x^2;
P[n_, x_] := P[n, x] = Expand[(42 + 66*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] = 1764 + 5544*x + 4356*x^2; for(n = 1, N, my(i = n + 1); v[i] = (42 + 66*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.