Real-rooted atlas

A046716 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(90 + 318\,x + 396\,x^{2} + 198\,x^{3} + 30\,x^{4} - 81\,n - 298\,n x - 384\,n x^{2} - 198\,n x^{3} - 31\,n x^{4} + 26\,n^{2} + 94\,n^{2} x + 120\,n^{2} x^{2} + 62\,n^{2} x^{3} + 10\,n^{2} x^{4} - 3\,n^{3} - 10\,n^{3} x - 12\,n^{3} x^{2} - 6\,n^{3} x^{3} - n^{3} x^{4}\right)\,P_{n-3}(x) + \left(24 + 102\,x + 162\,x^{2} + 114\,x^{3} + 30\,x^{4} - 23\,n - 100\,n x - 162\,n x^{2} - 116\,n x^{3} - 31\,n x^{4} + 8\,n^{2} + 34\,n^{2} x + 54\,n^{2} x^{2} + 38\,n^{2} x^{3} + 10\,n^{2} x^{4} - n^{3} - 4\,n^{3} x - 6\,n^{3} x^{2} - 4\,n^{3} x^{3} - n^{3} x^{4}\right)\,P_{n-3}'(x) + \left(-38 - 90\,x - 70\,x^{2} - 20\,x^{3} + 39\,n + 100\,n x + 83\,n x^{2} + 24\,n x^{3} - 12\,n^{2} - 33\,n^{2} x - 30\,n^{2} x^{2} - 9\,n^{2} x^{3} + n^{3} + 3\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-2}(x) + \left(-8 - 34\,x - 46\,x^{2} - 20\,x^{3} + 13\,n + 48\,n x + 59\,n x^{2} + 24\,n x^{3} - 6\,n^{2} - 21\,n^{2} x - 24\,n^{2} x^{2} - 9\,n^{2} x^{3} + n^{3} + 3\,n^{3} x + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(2 - 2\,x - 10\,x^{2} - 7\,n - 4\,n x + 7\,n x^{2} + 3\,n^{2} + 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8 - 18\,x - 10\,x^{2} + 5\,n + 12\,n x + 7\,n x^{2} - n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)
\(P_{3}(x) = 10 + 7\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3107118
44147141103
5209358191321791
61287306925878879917930
792702900134757195875009427198052
87622529878747271938223216348933770238611429609
97033373336516662141470772844352499151103626371016088123881885
1071907794020932196522654129322750105054261524081551537945423325661256911448545632
1180659818520905235147251484923837492342424235216159524577367181270717122283023012640111210319344470406
129847869377227149335235702804724488071664755077430687453106176552513388555291803980332071886033250364149109762061213698491707
13130000233051070067769223961910962378694262977481253544116774124287592748686078033449741357430206813360259205427092080666297670342111948132015320189732499
14184505557823168517312035769919171776891740242776386128887650446802336146186949382802614855684116792629396393712733900708220665826468343808412460813839191250914215426671143199647069878
152801704634214281396879477491294495195109893607568858981336786168683161059083171113025968879351868015266398369696169813383155279851571287506496514723389589416814657288499983935338013394471834834015514143097005511064
Technical details
Node id
n-d4ffae2f8973795e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8fdad8d4561f4147

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,        1;
   3:      10,        7,        1;
   4:      41,       47,       14,         1;
   5:     209,      358,      191,        32,         1;
   6:    1287,     3069,     2587,       887,        99,        1;
   7:    9270,    29001,    34757,     19587,      5009,      427,        1;
   8:   76225,   298787,   472719,    382232,    163489,    33770,     2386,       1;
   9:  703337,  3336516,  6621414,   7077284,   4352499,  1511036,   263710,   16088,      1;
  10: 7190779, 40209321, 96522654, 129322750, 105054261, 52408155, 15379454, 2332566, 125691, 1;
%F Formula
The row polynomials satisfy (2*n-4)*P(n,x) = -(3*n^3-26*n^2+81*n-90+(10*n^3-94*n^2+298*n-318)*x+(12*n^3-120*n^2+384*n-396)*x^2+(6*n^3-62*n^2+198*n-198)*x^3+(n^3-10*n^2+31*n-30)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(4*n^3-34*n^2+100*n-102)*x+(6*n^3-54*n^2+162*n-162)*x^2+(4*n^3-38*n^2+116*n-114)*x^3+(n^3-10*n^2+31*n-30)*x^4)*d/dx P(n-3,x) + (n^3-12*n^2+39*n-38+(3*n^3-33*n^2+100*n-90)*x+(3*n^3-30*n^2+83*n-70)*x^2+(n^3-9*n^2+24*n-20)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(3*n^3-21*n^2+48*n-34)*x+(3*n^3-24*n^2+59*n-46)*x^2+(n^3-9*n^2+24*n-20)*x^3)*d/dx P(n-2,x) + (3*n^2-7*n+2+(2*n^2-4*n-2)*x-(n^2-7*n+10)*x^2)*P(n-1,x) - (n^2-5*n+8+(2*n^2-12*n+18)*x+(n^2-7*n+10)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + x, P(3,x) = 10 + 7*x + x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[3, x_] := 10 + 7*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 - 26*n^2 + 81*n - 90 + (10*n^3 - 94*n^2 + 298*n - 318)*x + (12*n^3 - 120*n^2 + 384*n - 396)*x^2 + (6*n^3 - 62*n^2 + 198*n - 198)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (4*n^3 - 34*n^2 + 100*n - 102)*x + (6*n^3 - 54*n^2 + 162*n - 162)*x^2 + (4*n^3 - 38*n^2 + 116*n - 114)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*D[P[n - 3, x], x] + (n^3 - 12*n^2 + 39*n - 38 + (3*n^3 - 33*n^2 + 100*n - 90)*x + (3*n^3 - 30*n^2 + 83*n - 70)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (3*n^3 - 21*n^2 + 48*n - 34)*x + (3*n^3 - 24*n^2 + 59*n - 46)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*D[P[n - 2, x], x] + (3*n^2 - 7*n + 2 + (2*n^2 - 4*n - 2)*x - (n^2 - 7*n + 10)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (2*n^2 - 12*n + 18)*x + (n^2 - 7*n + 10)*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; v[3] = 10 + 7*x + x^2; for(n = 4, N, my(i = n); v[i] = (-(3*n^3 - 26*n^2 + 81*n - 90 + (10*n^3 - 94*n^2 + 298*n - 318)*x + (12*n^3 - 120*n^2 + 384*n - 396)*x^2 + (6*n^3 - 62*n^2 + 198*n - 198)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (4*n^3 - 34*n^2 + 100*n - 102)*x + (6*n^3 - 54*n^2 + 162*n - 162)*x^2 + (4*n^3 - 38*n^2 + 116*n - 114)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*deriv(v[i - 3], x) + (n^3 - 12*n^2 + 39*n - 38 + (3*n^3 - 33*n^2 + 100*n - 90)*x + (3*n^3 - 30*n^2 + 83*n - 70)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (3*n^3 - 21*n^2 + 48*n - 34)*x + (3*n^3 - 24*n^2 + 59*n - 46)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*deriv(v[i - 2], x) + (3*n^2 - 7*n + 2 + (2*n^2 - 4*n - 2)*x - (n^2 - 7*n + 10)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (2*n^2 - 12*n + 18)*x + (n^2 - 7*n + 10)*x^2)*deriv(v[i - 1], x))/(2*n - 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023