Real-rooted atlas

A038336 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(42 + 60\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1764 + 5040\,x + 3600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
017645040360010404
1740883175204536002160001061208
2311169617781120381024003628800012960000108243216
313069123293350880026671680003810240000272160000077760000011040808032
4548903174447048843520168031584000320060160000342921600000195955200000466560000001126162419264
523053933324823053933324809880257139200235244217600003360631680000028805414400000137168640000002799360000000114868566764928
69682651996416110658879959040553294399795200158084114227200028229306112000003226206412800000230443315200000094058496000000016796160000000011716593810022656
74066713838494725228632078064640298778975889408009959299196313600021341355420672000030487650600960000029035857715200000017777055744000000063489484800000000100776960000000001195092568622310912

All 15 rows

Technical details
Node id
n-1fba1ea01ed18c20 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-0fbe6fa5f58efdcf

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,                 5040,                 3600;
  1:              74088,               317520,               453600,                216000;
  2:            3111696,             17781120,             38102400,              36288000,              12960000;
  3:          130691232,            933508800,           2667168000,            3810240000,            2721600000,              777600000;
  4:         5489031744,          47048843520,         168031584000,          320060160000,          342921600000,           195955200000,            46656000000;
  5:       230539333248,        2305393332480,        9880257139200,        23524421760000,        33606316800000,         28805414400000,         13716864000000,          2799360000000;
  6:      9682651996416,      110658879959040,      553294399795200,      1580841142272000,      2822930611200000,       3226206412800000,       2304433152000000,        940584960000000,        167961600000000;
  7:    406671383849472,     5228632078064640,    29877897588940800,     99592991963136000,    213413554206720000,     304876506009600000,     290358577152000000,     177770557440000000,      63489484800000000,     10077696000000000;
  8:  17080198121677824,   244002830309683200,  1568589623419392000,   5975579517788160000,  14938948794470400000,   25609626504806400000,   30487650600960000000,   24887878041600000000,   13332791808000000000,   4232632320000000000,    604661760000000000;
  9: 717368321110468608, 11272930760307363840, 80520934002195456000, 345089717152266240000, 985970620435046400000, 1971941240870092800000, 2817058915528704000000, 2874549913804800000000, 2053249938432000000000, 977738065920000000000, 279353733120000000000, 36279705600000000000;
%F Formula
The row polynomials satisfy P(n,x) = (42+60*x)*P(n-1,x) for n >= 1, with P(0,x) = 1764 + 5040*x + 3600*x^2.
%t Mathematica
P[0, x_] := 1764 + 5040*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[(42 + 60*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 + 5040*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = (42 + 60*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.