Real-rooted atlas

A069466 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n^{2} + 2\,n^{3} x - 4\,n^{4} x + n^{6} x^{2}\right)\,P_{n}(x) = \left(16\,x + 8\,x^{2} - 2\,n - 56\,n x - 54\,n x^{2} + 4\,n^{2} + 44\,n^{2} x + 80\,n^{2} x^{2} + 8\,n^{2} x^{3} + 16\,n^{3} x + 34\,n^{3} x^{2} - 14\,n^{3} x^{3} - 16\,n^{4} x - 84\,n^{4} x^{2} - 16\,n^{4} x^{3} - 2\,n^{5} x^{2} + 18\,n^{5} x^{3} + 4\,n^{6} x^{2} + 12\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{3} + 8\,n x + 16\,n x^{2} - 24\,n x^{3} - 16\,n^{2} x^{2} + 8\,n^{2} x^{3} + 8\,n^{2} x^{4} - 32\,n^{3} x^{2} + 36\,n^{3} x^{3} - 4\,n^{3} x^{4} + 20\,n^{4} x^{3} - 20\,n^{4} x^{4} + 8\,n^{5} x^{3} - 8\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
269654156
32072016203202680
47044802268017920175046900
5252252002268004032001575009072822024
692413305618711005913600519750011975044527614358960
734326726721362160867267200105105000544864328240232219648249616224
812870329472090810720645765120157657500014529715204944139205271552010424704317601860
9486201575288056710368054890035201929727800027788080320168100732804032737280318995820486200074323935400
1018475673902400336717810042567782400203693490000422378820864399239240400170271129600303046029001847560000223554761273766246896
1170543234142908819205386200307286179200192053862000054205282010887377941162592491657886720015556362822002133931800001032822991210158220821741879825120
1227041561557593856106013731824209409840640016564645597500610639095306241131284311597441085580613877765366945173590013088115040000142529572785656073378816457002364369756112996816
1310400600703080560056949525360013611639641600132926168375000620180331170400150068327048640019600761083904001395405745133400531704673500000102938024789600911192405760029705153660020385176006267513512058400
144011660031451414400298985008140085044624537600100492183291500057883497575904001772682113262060030241174243737600293035206478014001607874932664000048638216713086007654016208384005614274041740015411193056009026235000105918519085560000
Technical details
Node id
n-3524d44278ffbb79 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-5622f61baf05ef75

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:     2,        8;
  2:     6,       96,        54;
  3:    20,      720,      1620,        320;
  4:    70,     4480,     22680,      17920,        1750;
  5:   252,    25200,    226800,     403200,      157500,        9072;
  6:   924,   133056,   1871100,    5913600,     5197500,     1197504,       45276;
  7:  3432,   672672,  13621608,   67267200,   105105000,    54486432,     8240232,     219648;
  8: 12870,  3294720,  90810720,  645765120,  1576575000,  1452971520,   494413920,   52715520,   1042470;
  9: 48620, 15752880, 567103680, 5489003520, 19297278000, 27788080320, 16810073280, 4032737280, 318995820, 4862000;
%F Formula
The row polynomials satisfy (n^2-(4*n^4-2*n^3)*x+n^6*x^2)*P(n,x) = (4*n^2-2*n-(16*n^4-16*n^3-44*n^2+56*n-16)*x+(4*n^6-2*n^5-84*n^4+34*n^3+80*n^2-54*n+8)*x^2+(12*n^6+18*n^5-16*n^4-14*n^3+8*n^2)*x^3)*P(n-1,x) + ((8*n-4)*x-(32*n^3+16*n^2-16*n)*x^2+(8*n^5+20*n^4+36*n^3+8*n^2-24*n+4)*x^3-(8*n^5+20*n^4+4*n^3-8*n^2)*x^4)*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[((4*n^2 - 2*n - (16*n^4 - 16*n^3 - 44*n^2 + 56*n - 16)*x + (4*n^6 - 2*n^5 - 84*n^4 + 34*n^3 + 80*n^2 - 54*n + 8)*x^2 + (12*n^6 + 18*n^5 - 16*n^4 - 14*n^3 + 8*n^2)*x^3)*P[n - 1, x] + ((8*n - 4)*x - (32*n^3 + 16*n^2 - 16*n)*x^2 + (8*n^5 + 20*n^4 + 36*n^3 + 8*n^2 - 24*n + 4)*x^3 - (8*n^5 + 20*n^4 + 4*n^3 - 8*n^2)*x^4)*D[P[n - 1, x], x])/(n^2 - (4*n^4 - 2*n^3)*x + n^6*x^2)]];
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] = ((4*n^2 - 2*n - (16*n^4 - 16*n^3 - 44*n^2 + 56*n - 16)*x + (4*n^6 - 2*n^5 - 84*n^4 + 34*n^3 + 80*n^2 - 54*n + 8)*x^2 + (12*n^6 + 18*n^5 - 16*n^4 - 14*n^3 + 8*n^2)*x^3)*v[i - 1] + ((8*n - 4)*x - (32*n^3 + 16*n^2 - 16*n)*x^2 + (8*n^5 + 20*n^4 + 36*n^3 + 8*n^2 - 24*n + 4)*x^3 - (8*n^5 + 20*n^4 + 4*n^3 - 8*n^2)*x^4)*deriv(v[i - 1], x))/(n^2 - (4*n^4 - 2*n^3)*x + n^6*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

Same fingerprint ?

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