Real-rooted atlas

A038333 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(14 + 19\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 532\,x + 361\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01965323611089
127441117215162685935937
2384162085444245363841041303211185921
353782436495209905840134436409122470247609939135393
4752953661311936208022640376421920383143740207992316470458811291467969
51054135041001428288407724374492223370401251602884010191623484461049633889387173942618442977
614757890561602285260876108549888206580349696350448807520380487276736258187794928100113634768169835630411406408618241
72066104678425235992857613699538979844338187343616883130994950411985349217184108438873869766307158990384213992894316632268769777946411484401953

All 15 rows

Technical details
Node id
n-155538ecb7326369 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b35a687a1de5c177

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:           196,            532,             361;
  1:          2744,          11172,           15162,             6859;
  2:         38416,         208544,          424536,           384104,           130321;
  3:        537824,        3649520,         9905840,         13443640,          9122470,          2476099;
  4:       7529536,       61311936,       208022640,        376421920,        383143740,        207992316,          47045881;
  5:     105413504,     1001428288,      4077243744,       9222337040,      12516028840,      10191623484,        4610496338,         893871739;
  6:    1475789056,    16022852608,     76108549888,     206580349696,     350448807520,     380487276736,      258187794928,      100113634768,      16983563041;
  7:   20661046784,   252359928576,   1369953897984,    4338187343616,    8831309949504,   11985349217184,    10843887386976,     6307158990384,    2139928943166,     322687697779;
  8:  289254654976,  3925598888960,  23974193214720,   86763746872320,  206063898821760,  335589778081152,   379536058544160,   294334086217920,  149795026021620,   45176277689060,   6131066257801;
  9: 4049565169664, 60454222889984, 410225083896320, 1670202127292160, 4533405774078720, 8613470970749568, 11689710603160128, 11331862319389920, 7689478002443160, 3478573382057620, 944184203701354, 116490258898219;
%F Formula
The row polynomials satisfy P(n,x) = (14+19*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 532*x + 361*x^2.
%t Mathematica
P[0, x_] := 196 + 532*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 19*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] = 196 + 532*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 19*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.