Real-rooted atlas

A063967 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(576 - 960\,x + 64\,x^{2} + 128\,x^{3} + 8\,x^{4} - 2\,x^{5} + 864\,n x - 504\,n x^{2} - 120\,n x^{3} + 26\,n x^{4} + 3\,n x^{5} + 312\,n^{2} x^{2} - 44\,n^{2} x^{3} - 29\,n^{2} x^{4} + 36\,n^{3} x^{3} + 2\,n^{3} x^{4} - n^{3} x^{5} + n^{4} x^{4}\right)\,P_{n}(x) = \left(1152 + 384\,x - 352\,x^{2} - 128\,x^{3} + 16\,x^{4} + 8\,x^{5} + 1728\,n x + 1104\,n x^{2} - 80\,n x^{3} - 144\,n x^{4} - 22\,n x^{5} + 624\,n^{2} x^{2} + 440\,n^{2} x^{3} + 30\,n^{2} x^{4} - 23\,n^{2} x^{5} - 3\,n^{2} x^{6} + 72\,n^{3} x^{3} + 48\,n^{3} x^{4} + 4\,n^{3} x^{5} - n^{3} x^{6} + 2\,n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-2}(x) + \left(1152 + 384\,x - 352\,x^{2} - 128\,x^{3} + 16\,x^{4} + 8\,x^{5} + 1728\,n x + 768\,n x^{2} - 352\,n x^{3} - 188\,n x^{4} - 18\,n x^{5} + 624\,n^{2} x^{2} + 368\,n^{2} x^{3} - 18\,n^{2} x^{4} - 27\,n^{2} x^{5} - 2\,n^{2} x^{6} + 72\,n^{3} x^{3} + 44\,n^{3} x^{4} + 2\,n^{3} x^{5} - n^{3} x^{6} + 2\,n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
3620935
416807216184
54428838717625920
612097617461200350364428
732831847110654429006124920727
88961011227072312321860059769806494932
9244831488982801363201020254413611025147281427275
10668896576344304557952503050273924922181875221061001895670
111827229267211733122175360229495015074286394011752962997029001218309682
124992087833639110408164608986680075918243887856134368030974445600387214436053424
131363842614784128034722971878440479700357114242143960789511042601882516400666715040169155045421
1437260877314564128854410548147215988580015910545610959545853670336188525884718600821106943206422196661624362
15101798422728448131463072366564096611981400678160080527237550296390720122164200369570008109904125524812979280362252804167755
Technical details
Node id
n-a618dbaa2ab6e2bf · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-48e60cd64bbf3ade

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    2,     4;
   3:    6,    20,      9;
   4:   16,    80,     72,     16;
   5:   44,   288,    387,    176,     25;
   6:  120,   976,   1746,   1200,    350,     36;
   7:  328,  3184,   7110,   6544,   2900,    612,    49;
   8:  896, 10112,  27072,  31232,  18600,   5976,   980,    64;
   9: 2448, 31488,  98280, 136320, 102025,  44136, 11025,  1472,   81;
  10: 6688, 96576, 344304, 557952, 503050, 273924, 92218, 18752, 2106, 100;
%F Formula
The row polynomials satisfy (576+(864*n-960)*x+(312*n^2-504*n+64)*x^2+(36*n^3-44*n^2-120*n+128)*x^3+(n^4+2*n^3-29*n^2+26*n+8)*x^4-(n^3-3*n+2)*x^5)*P(n,x) = (1152+(1728*n+384)*x+(624*n^2+1104*n-352)*x^2+(72*n^3+440*n^2-80*n-128)*x^3+(2*n^4+48*n^3+30*n^2-144*n+16)*x^4+(n^4+4*n^3-23*n^2-22*n+8)*x^5-(n^3+3*n^2)*x^6)*P(n-2,x) + (1152+(1728*n+384)*x+(624*n^2+768*n-352)*x^2+(72*n^3+368*n^2-352*n-128)*x^3+(2*n^4+44*n^3-18*n^2-188*n+16)*x^4+(n^4+2*n^3-27*n^2-18*n+8)*x^5-(n^3+2*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((1152 + (1728*n + 384)*x + (624*n^2 + 1104*n - 352)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (n^4 + 4*n^3 - 23*n^2 - 22*n + 8)*x^5 - (n^3 + 3*n^2)*x^6)*P[n - 2, x] + (1152 + (1728*n + 384)*x + (624*n^2 + 768*n - 352)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (n^4 + 2*n^3 - 27*n^2 - 18*n + 8)*x^5 - (n^3 + 2*n^2)*x^6)*P[n - 1, x])/(576 + (864*n - 960)*x + (312*n^2 - 504*n + 64)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (n^3 - 3*n + 2)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((1152 + (1728*n + 384)*x + (624*n^2 + 1104*n - 352)*x^2 + (72*n^3 + 440*n^2 - 80*n - 128)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (n^4 + 4*n^3 - 23*n^2 - 22*n + 8)*x^5 - (n^3 + 3*n^2)*x^6)*v[i - 2] + (1152 + (1728*n + 384)*x + (624*n^2 + 768*n - 352)*x^2 + (72*n^3 + 368*n^2 - 352*n - 128)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (n^4 + 2*n^3 - 27*n^2 - 18*n + 8)*x^5 - (n^3 + 2*n^2)*x^6)*v[i - 1])/(576 + (864*n - 960)*x + (312*n^2 - 504*n + 64)*x^2 + (36*n^3 - 44*n^2 - 120*n + 128)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (n^3 - 3*n + 2)*x^5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.