Real-rooted atlas

A090452 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(8\,x - 8\,x^{2} + 8\,x^{3} - 2\,n - 12\,n x + 6\,n x^{2} - 28\,n x^{3} - 3\,n^{2} - 6\,n^{2} x - 5\,n^{2} x^{2} + 16\,n^{2} x^{3} - n^{3} + 5\,n^{3} x + 15\,n^{3} x^{2} + 16\,n^{3} x^{3} + 4\,n^{4} x + 10\,n^{4} x^{2} + n^{5} x\right)\,P_{n}(x) = \left(-24\,x - 48\,x^{2} + 48\,x^{4} + 24\,x^{5} - 2\,n - 28\,n x - 122\,n x^{2} - 12\,n x^{3} + 180\,n x^{4} + 96\,n x^{5} - 3\,n^{2} - 28\,n^{2} x + 21\,n^{2} x^{2} + 94\,n^{2} x^{3} + 144\,n^{2} x^{4} + 96\,n^{2} x^{5} - n^{3} + 3\,n^{3} x + 34\,n^{3} x^{2} + 126\,n^{3} x^{3} + 96\,n^{3} x^{4} + 4\,n^{4} x + 24\,n^{4} x^{2} + 20\,n^{4} x^{3} + n^{5} x + n^{5} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} + 8\,x^{3} + 32\,x^{4} + 28\,x^{5} + 8\,x^{6} - 3\,n x - 16\,n x^{2} + 18\,n x^{3} + 104\,n x^{4} + 105\,n x^{5} + 32\,n x^{6} - 2\,n^{2} x + 10\,n^{2} x^{2} + 46\,n^{2} x^{3} + 86\,n^{2} x^{4} + 84\,n^{2} x^{5} + 32\,n^{2} x^{6} + 6\,n^{3} x^{2} + 32\,n^{3} x^{3} + 46\,n^{3} x^{4} + 20\,n^{3} x^{5} + 2\,n^{4} x^{2} + 4\,n^{4} x^{3} + 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11121831
2128144240125538
31484591680285022686867992
417210626336192753276031556161283402110592
511002070177608087521898837142039936026487099000159721470416
61132361841600259125989388247273641811844829625376200018926825559847250119060576
711685859862406956253498768117967992780755246873890569415004952917230126096121801682942940321750242806528

All 15 rows

Technical details
Node id
n-adf48e146cd7d0ad · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7149774de98d320a

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,  12,    18;
  2: 1,  28,   144,    240,     125;
  3: 1,  48,   459,   1680,    2850,     2268,       686;
  4: 1,  72,  1062,   6336,   19275,    32760,     31556,     16128,       3402;
  5: 1, 100,  2070,  17760,   80875,   218988,    371420,    399360,     264870,      99000,      15972;
  6: 1, 132,  3618,  41600,  259125,   989388,   2472736,   4181184,    4829625,    3762000,    1892682,     555984,      72501;
  7: 1, 168,  5859,  86240,  695625,  3498768,  11796799,  27807552,   46873890,   56941500,   49529172,   30126096,   12180168,    2942940,     321750;
  8: 1, 208,  8964, 163520, 1645700, 10435824,  45005716, 138322432,  311826915,  524106000,  660910712,  623278656,  433410978,  215815600,   72844200,   14936064,   1405118;
  9: 1, 252, 13122, 289536, 3538800, 27433728, 145986288, 560454912, 1603306467, 3489568500, 5849735562, 7592180544, 7617310974, 5855594472, 3386026800, 1425715200, 412721478, 73465704, 6063356;
%F Formula
The row polynomials satisfy -(n^3+3*n^2+2*n-(n^5+4*n^4+5*n^3-6*n^2-12*n+8)*x-(10*n^4+15*n^3-5*n^2+6*n-8)*x^2-(16*n^3+16*n^2-28*n+8)*x^3)*P(n,x) = -(n^3+3*n^2+2*n-(n^5+4*n^4+3*n^3-28*n^2-28*n-24)*x-(n^5+24*n^4+34*n^3+21*n^2-122*n-48)*x^2-(20*n^4+126*n^3+94*n^2-12*n)*x^3-(96*n^3+144*n^2+180*n+48)*x^4-(96*n^2+96*n+24)*x^5)*P(n-1,x) - ((2*n^2+3*n+4)*x-(2*n^4+6*n^3+10*n^2-16*n-8)*x^2-(4*n^4+32*n^3+46*n^2+18*n+8)*x^3-(2*n^4+46*n^3+86*n^2+104*n+32)*x^4-(20*n^3+84*n^2+105*n+28)*x^5-(32*n^2+32*n+8)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 2*n - (n^5 + 4*n^4 + 3*n^3 - 28*n^2 - 28*n - 24)*x - (n^5 + 24*n^4 + 34*n^3 + 21*n^2 - 122*n - 48)*x^2 - (20*n^4 + 126*n^3 + 94*n^2 - 12*n)*x^3 - (96*n^3 + 144*n^2 + 180*n + 48)*x^4 - (96*n^2 + 96*n + 24)*x^5)*P[n - 1, x] - ((2*n^2 + 3*n + 4)*x - (2*n^4 + 6*n^3 + 10*n^2 - 16*n - 8)*x^2 - (4*n^4 + 32*n^3 + 46*n^2 + 18*n + 8)*x^3 - (2*n^4 + 46*n^3 + 86*n^2 + 104*n + 32)*x^4 - (20*n^3 + 84*n^2 + 105*n + 28)*x^5 - (32*n^2 + 32*n + 8)*x^6)*D[P[n - 1, x], x])/(-n^3 - 3*n^2 - 2*n + (n^5 + 4*n^4 + 5*n^3 - 6*n^2 - 12*n + 8)*x + (10*n^4 + 15*n^3 - 5*n^2 + 6*n - 8)*x^2 + (16*n^3 + 16*n^2 - 28*n + 8)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 2*n - (n^5 + 4*n^4 + 3*n^3 - 28*n^2 - 28*n - 24)*x - (n^5 + 24*n^4 + 34*n^3 + 21*n^2 - 122*n - 48)*x^2 - (20*n^4 + 126*n^3 + 94*n^2 - 12*n)*x^3 - (96*n^3 + 144*n^2 + 180*n + 48)*x^4 - (96*n^2 + 96*n + 24)*x^5)*v[i - 1] - ((2*n^2 + 3*n + 4)*x - (2*n^4 + 6*n^3 + 10*n^2 - 16*n - 8)*x^2 - (4*n^4 + 32*n^3 + 46*n^2 + 18*n + 8)*x^3 - (2*n^4 + 46*n^3 + 86*n^2 + 104*n + 32)*x^4 - (20*n^3 + 84*n^2 + 105*n + 28)*x^5 - (32*n^2 + 32*n + 8)*x^6)*deriv(v[i - 1], x))/(-n^3 - 3*n^2 - 2*n + (n^5 + 4*n^4 + 5*n^3 - 6*n^2 - 12*n + 8)*x + (10*n^4 + 15*n^3 - 5*n^2 + 6*n - 8)*x^2 + (16*n^3 + 16*n^2 - 28*n + 8)*x^3)); 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. [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.