Real-rooted atlas

A178822 → basis A013611 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-240 + 160\,x - 108\,n + 72\,n x - 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-2}(x) + \left(-40 - 20\,x + 32\,n + 16\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 36 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1362460
275613443362436
3120963673623296358475712
4163296709632790272290304322561985760
51959552111121921838592011612160296755225804846295424
621555072150915072337193472315549696135776256264929281892352989374848
7221709312184767897652451942406664052736420588748813592494082141061121297612819770854400
8216166579220896432128722887925761176926699521004863242244714876108812145655808160255180884344832374507198208
92017554739222193204633690787605708818183433748481989989793792125581020364846550849945699451928576112834641925248122886790895727616
10181579926528223948883558410592085504000253213728768003417325053542427565240614912136609950597124168321597440759665786880755729694723148873728118740722580480
11158469754060821657542000640116386988949504324514410135552524613527273472521440321339392329075625295872133507665887232345322668687365482475421696485499076608183207198722013299340509184
1213469929095168202048961773568121678285989478438858366839685127356942675869696876419459789619268009619053608963499976965488640119783881808281626847182900428837737629483008301070496563210381741260833247377378295808
131119040263290881827765833564160121975366940098564397002958831616095757629760995328133784947384123392124310554587168768784572324627087363393506319217459210020867941597184197954250788044824954511609036818112145522688574988746752536621306229227520
149112185001082881609819369242624011795216622708326447427246633988915211708693520140206081885318322891784192205431248703297945615499661845337210888199529505992212483049700884972830727907613854190796813954853671796736159423844161945610612649440051231213674823688489357908845723648
Technical details
Node id
n-a81391ae04cc879c · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          36,           24;
  2:         756,         1344,          336;
  3:       12096,        36736,        23296,          3584;
  4:      163296,       709632,       790272,        290304,         32256;
  5:     1959552,     11112192,     18385920,      11612160,       2967552,        258048;
  6:    21555072,    150915072,    337193472,     315549696,     135776256,      26492928,      1892352;
  7:   221709312,   1847678976,   5245194240,    6664052736,    4205887488,    1359249408,    214106112,    12976128;
  8:  2161665792,  20896432128,  72288792576,  117692669952,  100486324224,   47148761088,  12145655808,  1602551808,    84344832;
  9: 20175547392, 221932046336, 907876057088, 1818343374848, 1989989793792, 1255810203648, 465508499456, 99451928576, 11283464192, 524812288;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(12*n^2+108*n+240-(8*n^2+72*n+160)*x)*P(n-2,x) + (8*n^2+32*n-40+(4*n^2+16*n-20)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 36 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 36 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n^2 + 108*n + 240 - (8*n^2 + 72*n + 160)*x)*P[n - 2, x] + (8*n^2 + 32*n - 40 + (4*n^2 + 16*n - 20)*x)*P[n - 1, x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 36 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^2 + 108*n + 240 - (8*n^2 + 72*n + 160)*x)*v[i - 2] + (8*n^2 + 32*n - 40 + (4*n^2 + 16*n - 20)*x)*v[i - 1])/(n^2 - n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

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.