Real-rooted atlas

A104684 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-16 + 42\,x - 45\,x^{2} + 12\,x^{3} - x^{4} + 16\,n - 82\,n x + 125\,n x^{2} - 50\,n x^{3} + 5\,n x^{4} + 40\,n^{2} x - 116\,n^{2} x^{2} + 76\,n^{2} x^{3} - 10\,n^{2} x^{4} + 36\,n^{3} x^{2} - 50\,n^{3} x^{3} + 10\,n^{3} x^{4} + 12\,n^{4} x^{3} - 5\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-176 + 366\,x + 105\,x^{2} - 56\,x^{3} + x^{4} + 112\,n - 766\,n x + 323\,n x^{2} + 278\,n x^{3} - 45\,n x^{4} - 4\,n x^{5} + 336\,n^{2} x - 728\,n^{2} x^{2} - 138\,n^{2} x^{3} + 149\,n^{2} x^{4} + 5\,n^{2} x^{5} + 300\,n^{3} x^{2} - 160\,n^{3} x^{3} - 129\,n^{3} x^{4} + 5\,n^{3} x^{5} + 92\,n^{4} x^{3} + 21\,n^{4} x^{4} - 9\,n^{4} x^{5} + 7\,n^{5} x^{4} + 3\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-16 + 72\,x + 96\,x^{2} - 12\,x^{3} - 20\,x^{4} - 96\,n x + 28\,n x^{2} + 182\,n x^{3} + 40\,n x^{4} - 20\,n x^{5} - 2\,n x^{6} - 120\,n^{2} x^{2} - 100\,n^{2} x^{3} + 64\,n^{2} x^{4} + 46\,n^{2} x^{5} + 2\,n^{2} x^{6} - 48\,n^{3} x^{3} - 68\,n^{3} x^{4} - 18\,n^{3} x^{5} + 2\,n^{3} x^{6} - 4\,n^{4} x^{4} - 6\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
31332954
46322813516442
532115361404384253670
61683100201242054408753630474
78989640321003056080016125172849252028
848639403396764127593040221375400683087642073796
926572925149445589360529356825585006612488780851208116976358
10146256315554628396796324438963226390700891777617122561751048019100138290410
11809745395600160275301585355345920251044500105090048266099403978240324405120001211121404372
12450467195845858921875702015274434600022474815001124023824354828012704985608486775566500173031449055583244
1325159596935597122561259279881220597723520191927216251117419494442349359361055020032169962705168960009423482419216972846528508
14140993361921599884644835194321001510374602881578189648751049033854444646280038413958630400283437168337945050031768308150508832955196583957620484
15792384825313067294620854827719373710862874597121258157225325940489437888477082799193168099689472413316073177007800800794674881569237762322229439042254666183972920
Technical details
Node id
n-59e41483dce6dc4f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-23befb70208d93ea

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:       3,        4;
   3:      13,       32,        9;
   4:      63,      228,      135,       16;
   5:     321,     1536,     1404,      384,       25;
   6:    1683,    10020,    12420,     5440,      875,      36;
   7:    8989,    64032,   100305,    60800,    16125,    1728,      49;
   8:   48639,   403396,   764127,   593040,   221375,   40068,    3087,     64;
   9:  265729,  2514944,  5589360,  5293568,  2558500,  661248,   87808,   5120,   81;
  10: 1462563, 15554628, 39679632, 44389632, 26390700, 8917776, 1712256, 175104, 8019, 100;
%F Formula
The row polynomials satisfy (16*n-16+(40*n^2-82*n+42)*x+(36*n^3-116*n^2+125*n-45)*x^2+(12*n^4-50*n^3+76*n^2-50*n+12)*x^3+(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x^4)*P(n,x) = (112*n-176+(336*n^2-766*n+366)*x+(300*n^3-728*n^2+323*n+105)*x^2+(92*n^4-160*n^3-138*n^2+278*n-56)*x^3+(7*n^5+21*n^4-129*n^3+149*n^2-45*n+1)*x^4+(3*n^5-9*n^4+5*n^3+5*n^2-4*n)*x^5)*P(n-1,x) - (16+(96*n-72)*x+(120*n^2-28*n-96)*x^2+(48*n^3+100*n^2-182*n+12)*x^3+(4*n^4+68*n^3-64*n^2-40*n+20)*x^4+(6*n^4+18*n^3-46*n^2+20*n)*x^5+(2*n^4-2*n^3-2*n^2+2*n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((112*n - 176 + (336*n^2 - 766*n + 366)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (92*n^4 - 160*n^3 - 138*n^2 + 278*n - 56)*x^3 + (7*n^5 + 21*n^4 - 129*n^3 + 149*n^2 - 45*n + 1)*x^4 + (3*n^5 - 9*n^4 + 5*n^3 + 5*n^2 - 4*n)*x^5)*P[n - 1, x] - (16 + (96*n - 72)*x + (120*n^2 - 28*n - 96)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (4*n^4 + 68*n^3 - 64*n^2 - 40*n + 20)*x^4 + (6*n^4 + 18*n^3 - 46*n^2 + 20*n)*x^5 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^6)*D[P[n - 1, x], x])/(16*n - 16 + (40*n^2 - 82*n + 42)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (12*n^4 - 50*n^3 + 76*n^2 - 50*n + 12)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((112*n - 176 + (336*n^2 - 766*n + 366)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (92*n^4 - 160*n^3 - 138*n^2 + 278*n - 56)*x^3 + (7*n^5 + 21*n^4 - 129*n^3 + 149*n^2 - 45*n + 1)*x^4 + (3*n^5 - 9*n^4 + 5*n^3 + 5*n^2 - 4*n)*x^5)*v[i - 1] - (16 + (96*n - 72)*x + (120*n^2 - 28*n - 96)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (4*n^4 + 68*n^3 - 64*n^2 - 40*n + 20)*x^4 + (6*n^4 + 18*n^3 - 46*n^2 + 20*n)*x^5 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^6)*deriv(v[i - 1], x))/(16*n - 16 + (40*n^2 - 82*n + 42)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (12*n^4 - 50*n^3 + 76*n^2 - 50*n + 12)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)); 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. [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