Real-rooted atlas

A126216 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(32 + 56\,x + 16\,x^{2} - 8\,x^{3} + 16\,n + 108\,n x + 80\,n x^{2} + 2\,n x^{3} + 40\,n^{2} x + 108\,n^{2} x^{2} + 41\,n^{2} x^{3} + 2\,n^{2} x^{4} + 36\,n^{3} x^{2} + 43\,n^{3} x^{3} + 5\,n^{3} x^{4} + 12\,n^{4} x^{3} + 4\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(32 + 56\,x + 16\,x^{2} - 8\,x^{3} + 112\,n + 412\,n x + 440\,n x^{2} + 146\,n x^{3} + 4\,n x^{4} + 2\,n x^{5} + 336\,n^{2} x + 780\,n^{2} x^{2} + 557\,n^{2} x^{3} + 128\,n^{2} x^{4} + 9\,n^{2} x^{5} + 300\,n^{3} x^{2} + 443\,n^{3} x^{3} + 181\,n^{3} x^{4} + 20\,n^{3} x^{5} + 92\,n^{4} x^{3} + 76\,n^{4} x^{4} + 14\,n^{4} x^{5} + 7\,n^{5} x^{4} + 3\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-16 - 96\,x - 196\,x^{2} - 184\,x^{3} - 86\,x^{4} - 20\,x^{5} - 2\,x^{6} - 96\,n x - 364\,n x^{2} - 512\,n x^{3} - 329\,n x^{4} - 94\,n x^{5} - 9\,n x^{6} - 120\,n^{2} x^{2} - 328\,n^{2} x^{3} - 314\,n^{2} x^{4} - 122\,n^{2} x^{5} - 16\,n^{2} x^{6} - 48\,n^{3} x^{3} - 92\,n^{3} x^{4} - 54\,n^{3} x^{5} - 10\,n^{3} x^{6} - 4\,n^{4} x^{4} - 6\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
21128948
34516810816337
4197968936288252414
59035500711033606253617534
642793109250355322409375118849128578
72079317550434171227664011025022176205864949197

All 15 rows

Technical details
Node id
n-9b47d19c23f3e144 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-0495df72540661b4

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:      3,       4;
  2:     11,      28,        9;
  3:     45,     168,      108,       16;
  4:    197,     968,      936,      288,       25;
  5:    903,    5500,     7110,     3360,      625,      36;
  6:   4279,   31092,    50355,    32240,     9375,    1188,     49;
  7:  20793,  175504,   341712,   276640,   110250,   22176,   2058,    64;
  8: 103049,  990736,  2253888,  2208640,  1122100,  312480,  46648,  3328,   81;
  9: 518859, 5597172, 14570604, 16770432, 10376100, 3710448, 773808, 89856, 5103, 100;
%F Formula
The row polynomials satisfy (16*n+32+(40*n^2+108*n+56)*x+(36*n^3+108*n^2+80*n+16)*x^2+(12*n^4+43*n^3+41*n^2+2*n-8)*x^3+(n^5+4*n^4+5*n^3+2*n^2)*x^4)*P(n,x) = (112*n+32+(336*n^2+412*n+56)*x+(300*n^3+780*n^2+440*n+16)*x^2+(92*n^4+443*n^3+557*n^2+146*n-8)*x^3+(7*n^5+76*n^4+181*n^3+128*n^2+4*n)*x^4+(3*n^5+14*n^4+20*n^3+9*n^2+2*n)*x^5)*P(n-1,x) - (16+(96*n+96)*x+(120*n^2+364*n+196)*x^2+(48*n^3+328*n^2+512*n+184)*x^3+(4*n^4+92*n^3+314*n^2+329*n+86)*x^4+(6*n^4+54*n^3+122*n^2+94*n+20)*x^5+(2*n^4+10*n^3+16*n^2+9*n+2)*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[((112*n + 32 + (336*n^2 + 412*n + 56)*x + (300*n^3 + 780*n^2 + 440*n + 16)*x^2 + (92*n^4 + 443*n^3 + 557*n^2 + 146*n - 8)*x^3 + (7*n^5 + 76*n^4 + 181*n^3 + 128*n^2 + 4*n)*x^4 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^5)*P[n - 1, x] - (16 + (96*n + 96)*x + (120*n^2 + 364*n + 196)*x^2 + (48*n^3 + 328*n^2 + 512*n + 184)*x^3 + (4*n^4 + 92*n^3 + 314*n^2 + 329*n + 86)*x^4 + (6*n^4 + 54*n^3 + 122*n^2 + 94*n + 20)*x^5 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^6)*D[P[n - 1, x], x])/(16*n + 32 + (40*n^2 + 108*n + 56)*x + (36*n^3 + 108*n^2 + 80*n + 16)*x^2 + (12*n^4 + 43*n^3 + 41*n^2 + 2*n - 8)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^4)]];
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] = ((112*n + 32 + (336*n^2 + 412*n + 56)*x + (300*n^3 + 780*n^2 + 440*n + 16)*x^2 + (92*n^4 + 443*n^3 + 557*n^2 + 146*n - 8)*x^3 + (7*n^5 + 76*n^4 + 181*n^3 + 128*n^2 + 4*n)*x^4 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^5)*v[i - 1] - (16 + (96*n + 96)*x + (120*n^2 + 364*n + 196)*x^2 + (48*n^3 + 328*n^2 + 512*n + 184)*x^3 + (4*n^4 + 92*n^3 + 314*n^2 + 329*n + 86)*x^4 + (6*n^4 + 54*n^3 + 122*n^2 + 94*n + 20)*x^5 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^6)*deriv(v[i - 1], x))/(16*n + 32 + (40*n^2 + 108*n + 56)*x + (36*n^3 + 108*n^2 + 80*n + 16)*x^2 + (12*n^4 + 43*n^3 + 41*n^2 + 2*n - 8)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^4)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023