Real-rooted atlas

A112007 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(-4\,x + 8\,n x + n^{2} - 7\,n^{2} x^{2} - 2\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(x - 5\,x^{2} + 10\,n x^{2} - 3\,n^{2} x + 4\,n^{2} x^{2} - 13\,n^{2} x^{3} - 2\,n^{3} x + 4\,n^{3} x^{2} - 14\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(x^{2} - x^{3} - 2\,n x^{2} + 2\,n x^{3} - n^{2} x^{2} + 4\,n^{2} x^{3} - 3\,n^{2} x^{4} + 4\,n^{3} x^{3} - 4\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-n + n^{2} + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-x + 2\,n x + 2\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
2632947
32423219816470
412017762952832255705
572014832396002323228503681270
650401359365232605139201402508640491327095
74032013645447067736103043204895000718200242066424414390
836288014935680992366641990410241441375003825000032957406425681499323825
9362880017735616014596822083822478464388393460016115400002609818401400627216410610011233772550
1039916800227342592022567330944742746232321000673772005934183494415404116280161560985656298078407200121275640940575
114790016003131087616036709471641614751966435842529510825600201749035622476189907841613129489766493049864562170626009880861447323799432950
12622702080046134434304062813649312003012927458918463746636835600654547911229443369828227281685966988943361026255239856507259276008106632162355264169209452414196025
13871782912007243970227200112958136161280635033248892928161820937849680020697974447253121386695593239488488629257561088878587975056967501605222800264969406768295315027255327221966414282538413750
1413076743680001207855079424002132174443887360138403720515133444166061215678640064698163944860160544756261559143202523939762019328063497021713362728308444327968005200894676714013381238764801054693341012838784225209402343790014375
Technical details
Node id
n-6bbfb631a2c1156c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3d72d99fe3345b32

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:       2,         4;
  2:       6,        32,          9;
  3:      24,       232,        198,         16;
  4:     120,      1776,       2952,        832,         25;
  5:     720,     14832,      39600,      23232,       2850,         36;
  6:    5040,    135936,     523260,     513920,     140250,       8640,        49;
  7:   40320,   1364544,    7067736,   10304320,    4895000,     718200,     24206,       64;
  8:  362880,  14935680,   99236664,  199041024,  144137500,   38250000,   3295740,    64256,     81;
  9: 3628800, 177356160, 1459682208, 3822478464, 3883934600, 1611540000, 260981840, 14006272, 164106, 100;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(n^4-n^2+(4*n^4-8*n+4)*x-(9*n^4-2*n^3-7*n^2)*x^2)*P(n-2,x) - ((2*n^3+3*n^2-1)*x-(4*n^4+4*n^3+4*n^2+10*n-5)*x^2-(4*n^4-14*n^3-13*n^2)*x^3)*d/dx P(n-2,x) - ((n^2+2*n-1)*x^2-(4*n^3+4*n^2+2*n-1)*x^3+(4*n^3+3*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3+n^2-n+4*n^3*x)*P(n-1,x) + ((2*n^2+2*n-1)*x-(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - n^2 + (4*n^4 - 8*n + 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*P[n - 2, x] - ((2*n^3 + 3*n^2 - 1)*x - (4*n^4 + 4*n^3 + 4*n^2 + 10*n - 5)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*D[P[n - 2, x], x] - ((n^2 + 2*n - 1)*x^2 - (4*n^3 + 4*n^2 + 2*n - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 + n^2 - n + 4*n^3*x)*P[n - 1, x] + ((2*n^2 + 2*n - 1)*x - (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 - n^2 + (4*n^4 - 8*n + 4)*x - (9*n^4 - 2*n^3 - 7*n^2)*x^2)*v[i - 2] - ((2*n^3 + 3*n^2 - 1)*x - (4*n^4 + 4*n^3 + 4*n^2 + 10*n - 5)*x^2 - (4*n^4 - 14*n^3 - 13*n^2)*x^3)*deriv(v[i - 2], x) - ((n^2 + 2*n - 1)*x^2 - (4*n^3 + 4*n^2 + 2*n - 1)*x^3 + (4*n^3 + 3*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 + n^2 - n + 4*n^3*x)*v[i - 1] + ((2*n^2 + 2*n - 1)*x - (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^2)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.