Real-rooted atlas

A116414 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n - 22\,n x - 16\,n x^{2} + 8\,n x^{3} - 20\,n^{2} + 8\,n^{2} x + 16\,n^{2} x^{2} + 16\,n^{3} + 8\,n^{3} x\right)\,P_{n}(x) = \left(40\,x^{2} + 28\,x^{3} - 16\,x^{4} - 16\,x^{5} - 2\,n - 58\,n x - 184\,n x^{2} - 116\,n x^{3} + 16\,n x^{4} + 16\,n x^{5} - 4\,n^{2} - 68\,n^{2} x + 40\,n^{2} x^{2} + 112\,n^{2} x^{3} + 32\,n^{2} x^{4} + 16\,n^{3} + 104\,n^{3} x + 80\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 20\,x^{3} - 24\,x^{4} - 48\,x^{5} - 16\,x^{6} - 2\,n x - 28\,n x^{2} - 112\,n x^{3} - 112\,n x^{4} - 32\,n x^{5} - 8\,n^{2} x - 68\,n^{2} x^{2} - 64\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
29341356
3139616240311
4171906396681211635
5213161636342025513648308
625474334510856161349294109341221
7296645958265606149070128328043280200913
8338869667551401761353130522878741131769841965804
937114014664102228419601103138814761801133016383817295244591595
1041142621141174480878556278245855124796570984431797512843148857321632427
114517442929027957616719246513024166526642747532027966999160428964251526265720101150728
124920943930342622029560051370930443193220920495521296013501148409705839210513950324797161469957657
1353247651372624140492959526571564998485682625217444775484065846997204579691442089559884543820323914842171552457
1457289065689884088783910648213286210920985662266032148734371423525185082542783846178240312073717513214707629471744539986667200
Technical details
Node id
n-640e334a90295685 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-602803251b9c8a8c

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:  5,    4;
  2:  9,   34,    13;
  3: 13,   96,   162,     40;
  4: 17,  190,   639,    668,    121;
  5: 21,  316,  1636,   3420,   2551,     364;
  6: 25,  474,  3345,  10856,  16134,    9294,    1093;
  7: 29,  664,  5958,  26560,  61490,   70128,   32804,    3280;
  8: 33,  886,  9667,  55140, 176135,  313052,  287874,  113176,   9841;
  9: 37, 1140, 14664, 102228, 419601, 1031388, 1476180, 1133016, 383817, 29524;
%F Formula
The row polynomials satisfy (16*n^3-20*n^2+6*n+(8*n^3+8*n^2-22*n)*x+(16*n^2-16*n)*x^2+8*n*x^3)*P(n,x) = (16*n^3-4*n^2-2*n+(104*n^3-68*n^2-58*n)*x+(80*n^3+40*n^2-184*n+40)*x^2+(16*n^3+112*n^2-116*n+28)*x^3+(32*n^2+16*n-16)*x^4+(16*n-16)*x^5)*P(n-1,x) - ((8*n^2+2*n)*x+(68*n^2+28*n-3)*x^2+(64*n^2+112*n-20)*x^3+(16*n^2+112*n+24)*x^4+(32*n+48)*x^5+16*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[((16*n^3 - 4*n^2 - 2*n + (104*n^3 - 68*n^2 - 58*n)*x + (80*n^3 + 40*n^2 - 184*n + 40)*x^2 + (16*n^3 + 112*n^2 - 116*n + 28)*x^3 + (32*n^2 + 16*n - 16)*x^4 + (16*n - 16)*x^5)*P[n - 1, x] - ((8*n^2 + 2*n)*x + (68*n^2 + 28*n - 3)*x^2 + (64*n^2 + 112*n - 20)*x^3 + (16*n^2 + 112*n + 24)*x^4 + (32*n + 48)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(16*n^3 - 20*n^2 + 6*n + (8*n^3 + 8*n^2 - 22*n)*x + (16*n^2 - 16*n)*x^2 + 8*n*x^3)]];
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 - 4*n^2 - 2*n + (104*n^3 - 68*n^2 - 58*n)*x + (80*n^3 + 40*n^2 - 184*n + 40)*x^2 + (16*n^3 + 112*n^2 - 116*n + 28)*x^3 + (32*n^2 + 16*n - 16)*x^4 + (16*n - 16)*x^5)*v[i - 1] - ((8*n^2 + 2*n)*x + (68*n^2 + 28*n - 3)*x^2 + (64*n^2 + 112*n - 20)*x^3 + (16*n^2 + 112*n + 24)*x^4 + (32*n + 48)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(16*n^3 - 20*n^2 + 6*n + (8*n^3 + 8*n^2 - 22*n)*x + (16*n^2 - 16*n)*x^2 + 8*n*x^3)); 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. [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