Real-rooted atlas

A038319 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 30\,x + 15\,n x\right)\,P_{n}(x) = \left(256 + 960\,x + 675\,x^{2} + 240\,n x + 225\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 960\,x + 675\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02569606751891
1409623040324001350073036
26553649152010368008640002531252710981
310485769830400276480003456000020250000455625097893226
4167772161887436806635520001105920000972000000437400000797343753464127271
526843545635232153601486356480030965760000362880000002449440000089302500001366875000120700500616
642949672966442450944031708938240079272345600011612160000001045094400000571536000000174960000000230660156254154404730761
7687194767361159641169920652298158080019025362944000334430208000003762339840000027433728000000125971200000003321506250000384433593750141579912215206

All 15 rows

Technical details
Node id
n-4d9577c2550aa313 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8a470edf89a00b77

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:            256,             960,              675;
  1:           4096,           23040,            32400,            13500;
  2:          65536,          491520,          1036800,           864000,            253125;
  3:        1048576,         9830400,         27648000,         34560000,          20250000,           4556250;
  4:       16777216,       188743680,        663552000,       1105920000,         972000000,         437400000,          79734375;
  5:      268435456,      3523215360,      14863564800,      30965760000,       36288000000,       24494400000,        8930250000,        1366875000;
  6:     4294967296,     64424509440,     317089382400,     792723456000,     1161216000000,     1045094400000,      571536000000,      174960000000,       23066015625;
  7:    68719476736,   1159641169920,    6522981580800,   19025362944000,    33443020800000,    37623398400000,    27433728000000,    12597120000000,     3321506250000,     384433593750;
  8:  1099511627776,  20615843020800,  130459631616000,  434865438720000,   891813888000000,  1203948748800000,  1097349120000000,   671846400000000,   265720500000000,   61509375000000,    6343154296875;
  9: 17592186044416, 362838837166080, 2551210573824000, 9567039651840000, 22422749184000000, 35315829964800000, 38626689024000000, 29561241600000000, 15588936000000000, 5412825000000000, 1116395156250000, 103797070312500;
%F Formula
The row polynomials satisfy (16+(15*n+30)*x)*P(n,x) = (256+(240*n+960)*x+(225*n+675)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 960*x + 675*x^2.
%t Mathematica
P[0, x_] := 256 + 960*x + 675*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((256 + (240*n + 960)*x + (225*n + 675)*x^2)*P[n - 1, x])/(16 + (15*n + 30)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 960*x + 675*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (240*n + 960)*x + (225*n + 675)*x^2)*v[i - 1])/(16 + (15*n + 30)*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

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.