Real-rooted atlas

A008517 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(8\,x + 14\,x^{2} + 8\,n x - 6\,n x^{2} - 16\,n^{2} x - 80\,n^{2} x^{2} + 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 2\,x^{2} + 16\,x^{3} - 4\,n x + 8\,n x^{2} + 4\,n x^{3} + 16\,n^{2} x^{2} - 88\,n^{2} x^{3} + 32\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 2\,x^{3} + 3\,x^{4} - 2\,n x^{2} + 2\,n x^{4} + 16\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-1 + 2\,n - 8\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x^{2} + 4\,n x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
21325487
3188522384995
4120829527104300013265
5145613068704009270025920202545
6196050490513920145350012234242469603489255
7119761795503132800161005002827094416715664258048066981915
81401660534017000000144137500447842304540288504238970880293932801418241825
918104196963285218560111912500055928658241170634029610379962368359146224036288000032839832025
101162886255342403902464785924300059341834944196132059312297098492928203105978496568356480004829932800825613363575
111326641953563418380220165128706940055976258822427460285404166475547713536746818300814440788301824009471540038406897623040022397626926675
1216542460297264811614841631674544440048356431280643369828227281611636407310745620653910334734418830796618240084449461852800166083963494401052366515200652180214711025
13113095218457189235037276928187540130570039048354446976374106775320208181119424423116846570442506319526472837513987200480243894475276818073301785804803060577420992001708694507520020289055572410625
1412620165616710101486804307201074565015850029910399580684838411778567342882523939762019328090051545278144080179717122069056000201637362838846176124563348463620096400374978907737605918489889177600294226732800000671610167527674375
Technical details
Node id
n-a6a9ef684f8dadd1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3c9a530453f07c86

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: 1,    8;
  2: 1,   32,      54;
  3: 1,   88,     522,      384;
  4: 1,  208,    2952,     7104,       3000;
  5: 1,  456,   13068,    70400,      92700,      25920;
  6: 1,  960,   50490,   513920,    1453500,    1223424,      246960;
  7: 1, 1976,  179550,  3132800,   16100500,   28270944,    16715664,     2580480;
  8: 1, 4016,  605340, 17000000,  144137500,  447842304,   540288504,   238970880,   29393280;
  9: 1, 8104, 1969632, 85218560, 1119125000, 5592865824, 11706340296, 10379962368, 3591462240, 362880000;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = -((16*n^2-8*n-8)*x-(72*n^3-80*n^2-6*n+14)*x^2)*P(n-2,x) - ((4*n+2)*x-(16*n^2+8*n+2)*x^2-(32*n^3-88*n^2+4*n+16)*x^3)*d/dx P(n-2,x) - ((2*n+1)*x^2-(16*n^2-2)*x^3+(16*n^2-2*n-3)*x^4)*d^2/dx^2 P(n-2,x) + (2*n-1+(16*n^2-8*n)*x)*P(n-1,x) + (4*n*x-(16*n^2-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((16*n^2 - 8*n - 8)*x - (72*n^3 - 80*n^2 - 6*n + 14)*x^2)*P[n - 2, x] - ((4*n + 2)*x - (16*n^2 + 8*n + 2)*x^2 - (32*n^3 - 88*n^2 + 4*n + 16)*x^3)*D[P[n - 2, x], x] - ((2*n + 1)*x^2 - (16*n^2 - 2)*x^3 + (16*n^2 - 2*n - 3)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n - 1 + (16*n^2 - 8*n)*x)*P[n - 1, x] + (4*n*x - (16*n^2 - 4)*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-((16*n^2 - 8*n - 8)*x - (72*n^3 - 80*n^2 - 6*n + 14)*x^2)*v[i - 2] - ((4*n + 2)*x - (16*n^2 + 8*n + 2)*x^2 - (32*n^3 - 88*n^2 + 4*n + 16)*x^3)*deriv(v[i - 2], x) - ((2*n + 1)*x^2 - (16*n^2 - 2)*x^3 + (16*n^2 - 2*n - 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n - 1 + (16*n^2 - 8*n)*x)*v[i - 1] + (4*n*x - (16*n^2 - 4)*x^2)*deriv(v[i - 1], x))/(2*n - 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. [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.