Real-rooted atlas

A091044 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - 2\,n x + 4\,n^{2} - 4\,n^{2} x + 4\,n^{3} + 12\,n^{3} x + 4\,n^{3} x^{2} + 24\,n^{4} x + 16\,n^{4} x^{2} + 16\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-16\,x - 8\,x^{2} - 16\,x^{3} + 4\,n - 24\,n x - 176\,n x^{2} - 80\,n x^{3} + 16\,n^{2} + 112\,n^{2} x - 8\,n^{2} x^{2} - 80\,n^{2} x^{3} + 16\,n^{3} + 240\,n^{3} x + 688\,n^{3} x^{2} + 496\,n^{3} x^{3} + 96\,n^{4} x + 544\,n^{4} x^{2} + 640\,n^{4} x^{3} + 64\,n^{5} x^{2} + 192\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x - 12\,x^{2} - 24\,x^{3} - 32\,x^{4} - 8\,n x - 80\,n x^{2} - 248\,n x^{3} - 224\,n x^{4} - 8\,n^{2} x - 128\,n^{2} x^{2} - 536\,n^{2} x^{3} - 608\,n^{2} x^{4} - 48\,n^{3} x^{2} - 320\,n^{3} x^{3} - 512\,n^{3} x^{4} - 32\,n^{4} x^{3} - 128\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
143236
216256432704
36415365760409611456
42568192483848192032000170752
51024409603317769175048960002211842408448
6409619660820275207864320129024008257536140492832657408
7163849175041150156857671680135168000148635648674365448388608429735936

All 15 rows

Technical details
Node id
n-fe140415adab6c9f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ae96443b231c3935

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:      4,       32;
  2:     16,      256,       432;
  3:     64,     1536,      5760,       4096;
  4:    256,     8192,     48384,      81920,      32000;
  5:   1024,    40960,    331776,     917504,     896000,      221184;
  6:   4096,   196608,   2027520,    7864320,   12902400,     8257536,     1404928;
  7:  16384,   917504,  11501568,   57671680,  135168000,   148635648,    67436544,     8388608;
  8:  65536,  4194304,  61931520,  381681664, 1171456000,  1868562432,  1483603968,   503316480,   47775744;
  9: 262144, 18874368, 320864256, 2348810240, 8945664000, 18893242368, 22042116096, 13287555072, 3503554560, 262144000;
%F Formula
The row polynomials satisfy (4*n^3+4*n^2+n+(24*n^4+12*n^3-4*n^2-2*n)*x+(16*n^5+16*n^4+4*n^3)*x^2)*P(n,x) = (16*n^3+16*n^2+4*n+(96*n^4+240*n^3+112*n^2-24*n-16)*x+(64*n^5+544*n^4+688*n^3-8*n^2-176*n-8)*x^2+(192*n^5+640*n^4+496*n^3-80*n^2-80*n-16)*x^3)*P(n-1,x) - ((8*n^2+8*n+2)*x+(48*n^3+128*n^2+80*n+12)*x^2+(32*n^4+320*n^3+536*n^2+248*n+24)*x^3+(128*n^4+512*n^3+608*n^2+224*n+32)*x^4)*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[((16*n^3 + 16*n^2 + 4*n + (96*n^4 + 240*n^3 + 112*n^2 - 24*n - 16)*x + (64*n^5 + 544*n^4 + 688*n^3 - 8*n^2 - 176*n - 8)*x^2 + (192*n^5 + 640*n^4 + 496*n^3 - 80*n^2 - 80*n - 16)*x^3)*P[n - 1, x] - ((8*n^2 + 8*n + 2)*x + (48*n^3 + 128*n^2 + 80*n + 12)*x^2 + (32*n^4 + 320*n^3 + 536*n^2 + 248*n + 24)*x^3 + (128*n^4 + 512*n^3 + 608*n^2 + 224*n + 32)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 4*n^2 + n + (24*n^4 + 12*n^3 - 4*n^2 - 2*n)*x + (16*n^5 + 16*n^4 + 4*n^3)*x^2)]];
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] = ((16*n^3 + 16*n^2 + 4*n + (96*n^4 + 240*n^3 + 112*n^2 - 24*n - 16)*x + (64*n^5 + 544*n^4 + 688*n^3 - 8*n^2 - 176*n - 8)*x^2 + (192*n^5 + 640*n^4 + 496*n^3 - 80*n^2 - 80*n - 16)*x^3)*v[i - 1] - ((8*n^2 + 8*n + 2)*x + (48*n^3 + 128*n^2 + 80*n + 12)*x^2 + (32*n^4 + 320*n^3 + 536*n^2 + 248*n + 24)*x^3 + (128*n^4 + 512*n^3 + 608*n^2 + 224*n + 32)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 4*n^2 + n + (24*n^4 + 12*n^3 - 4*n^2 - 2*n)*x + (16*n^5 + 16*n^4 + 4*n^3)*x^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. [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