Real-rooted atlas

A103327 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(3 + 24\,x + 27\,x^{2} - 5\,n - 67\,n x - 135\,n x^{2} + 2\,n^{2} + 67\,n^{2} x + 279\,n^{2} x^{2} - 28\,n^{3} x - 305\,n^{3} x^{2} + 4\,n^{4} x + 186\,n^{4} x^{2} - 60\,n^{5} x^{2} + 8\,n^{6} x^{2}\right)\,P_{n}(x) = \left(16\,x + 28\,x^{2} + 4\,x^{3} + n - 44\,n x - 111\,n x^{2} - 22\,n x^{3} + 2\,n^{2} + 31\,n^{2} x + 165\,n^{2} x^{2} + 30\,n^{2} x^{3} - 16\,n^{3} x - 111\,n^{3} x^{2} - 93\,n^{3} x^{3} + 4\,n^{4} x + 66\,n^{4} x^{2} + 162\,n^{4} x^{3} - 36\,n^{5} x^{2} - 108\,n^{5} x^{3} + 8\,n^{6} x^{2} + 24\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} + 6\,x^{3} + 2\,x^{4} + 4\,n x + 20\,n x^{2} - 14\,n x^{3} - 10\,n x^{4} - 28\,n^{2} x^{2} + 18\,n^{2} x^{3} + 10\,n^{2} x^{4} + 8\,n^{3} x^{2} + 20\,n^{3} x^{3} - 28\,n^{3} x^{4} - 40\,n^{4} x^{3} + 40\,n^{4} x^{4} + 16\,n^{5} x^{3} - 16\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
210818
33516827230
4841008972642128
516536968910352012516416
62861029646332457609750216112640
74552402417374532032017062522680343712192
86804950452509615558401547000514080466485124239360
996993024136047659121929447750586051213294688755272924092672
10133016279231395601881152044089500439538401861255230643201530901000131989504
1117712691926619239523001601690097502471182561681777025168486464552952530001331701890560
12230042504012978900130750400557175000112326480011211846805537664001291059001265000039930017283641704448
132925645840239768102999568001629736875433259280059626639804319377920161843467529601000023359050606528219718507366400
14365495004042141060640960320432466125014658605640266026546802657071872014601877290429214500063225162041041728891982274492408905728
Technical details
Node id
n-a66bec3b2cf26af5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b7e4e4f9193dc8a9

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   1;
  2:  10,     8;
  3:  35,   168,      27;
  4:  84,  1008,     972,      64;
  5: 165,  3696,    8910,    3520,     125;
  6: 286, 10296,   46332,   45760,    9750,     216;
  7: 455, 24024,  173745,  320320,  170625,   22680,     343;
  8: 680, 49504,  525096, 1555840, 1547000,  514080,   46648,   512;
  9: 969, 93024, 1360476, 5912192, 9447750, 5860512, 1329468, 87552, 729;
%F Formula
The row polynomials satisfy (2*n^2-5*n+3+(4*n^4-28*n^3+67*n^2-67*n+24)*x+(8*n^6-60*n^5+186*n^4-305*n^3+279*n^2-135*n+27)*x^2)*P(n,x) = (2*n^2+n+(4*n^4-16*n^3+31*n^2-44*n+16)*x+(8*n^6-36*n^5+66*n^4-111*n^3+165*n^2-111*n+28)*x^2+(24*n^6-108*n^5+162*n^4-93*n^3+30*n^2-22*n+4)*x^3)*P(n-1,x) + ((4*n-2)*x+(8*n^3-28*n^2+20*n-6)*x^2+(16*n^5-40*n^4+20*n^3+18*n^2-14*n+6)*x^3-(16*n^5-40*n^4+28*n^3-10*n^2+10*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (4*n^4 - 16*n^3 + 31*n^2 - 44*n + 16)*x + (8*n^6 - 36*n^5 + 66*n^4 - 111*n^3 + 165*n^2 - 111*n + 28)*x^2 + (24*n^6 - 108*n^5 + 162*n^4 - 93*n^3 + 30*n^2 - 22*n + 4)*x^3)*P[n - 1, x] + ((4*n - 2)*x + (8*n^3 - 28*n^2 + 20*n - 6)*x^2 + (16*n^5 - 40*n^4 + 20*n^3 + 18*n^2 - 14*n + 6)*x^3 - (16*n^5 - 40*n^4 + 28*n^3 - 10*n^2 + 10*n - 2)*x^4)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3 + (4*n^4 - 28*n^3 + 67*n^2 - 67*n + 24)*x + (8*n^6 - 60*n^5 + 186*n^4 - 305*n^3 + 279*n^2 - 135*n + 27)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + n + (4*n^4 - 16*n^3 + 31*n^2 - 44*n + 16)*x + (8*n^6 - 36*n^5 + 66*n^4 - 111*n^3 + 165*n^2 - 111*n + 28)*x^2 + (24*n^6 - 108*n^5 + 162*n^4 - 93*n^3 + 30*n^2 - 22*n + 4)*x^3)*v[i - 1] + ((4*n - 2)*x + (8*n^3 - 28*n^2 + 20*n - 6)*x^2 + (16*n^5 - 40*n^4 + 20*n^3 + 18*n^2 - 14*n + 6)*x^3 - (16*n^5 - 40*n^4 + 28*n^3 - 10*n^2 + 10*n - 2)*x^4)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3 + (4*n^4 - 28*n^3 + 67*n^2 - 67*n + 24)*x + (8*n^6 - 60*n^5 + 186*n^4 - 305*n^3 + 279*n^2 - 135*n + 27)*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

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