Real-rooted atlas

A008955 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-n^{2} - 4\,n^{2} x + 9\,n^{2} x^{2} + 8\,n^{3} x - 36\,n^{3} x^{2} - 4\,n^{4} x + 54\,n^{4} x^{2} - 36\,n^{5} x^{2} + 9\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(4\,n^{2} x^{2} + 4\,n^{2} x^{3} - 8\,n^{3} x^{2} - 16\,n^{3} x^{3} + 4\,n^{4} x^{2} + 24\,n^{4} x^{3} - 16\,n^{5} x^{3} + 4\,n^{6} x^{3}\right)\,P_{n-2}'(x) + \left(1 - 2\,n + 2\,n^{2} + 4\,n^{2} x - 8\,n^{3} x + 4\,n^{4} x\right)\,P_{n-1}(x) + \left(-4\,n^{2} x^{2} + 8\,n^{3} x^{2} - 4\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
21203657
31564415761074
411202457131201440030098
5122092071223205269005184001177048
61364270277115687407400278328962540160061380856
7156067158306592061886825550499040188170113616257024004122923040
8181614779810707008368478825625398883249836284176158920556544131681894400347072058400
911140296514319899201723584525492333589807393385529045431396008960164235220738561316818944000035813435306800
10115405530148470352067220095252974295305807440545747404101998064143360703834579457856204076350720000015933509222400004447691855586800
111202497227920366297622736268775146867247021656425537047649127790798797798416323881871162096107181484636089600300381536011392000229442532802560000654665260459605600
12126001628055452566400685604383756183265163400344285341209985118905151192256002492226126801496803009204928398240000189756834190036439045170639733224243200038775788043632640000112728537388697340000
1313276261670594170648018806625337522868133446340176660808615763587886234517739840279249232310326278055007503129565272000634327999227159741504386817112200061167206410294236997808744448000760005445655199744000022454680246940929760000
1414060406142118534828004764638628757594795279890078673179088978955401379045740944002459377137316010184073072169489282392000013679907441434859649024151829108726854797987840900081044255782779346944234762022314188174131200017100122527241994240000005123978399838068343520000
Technical details
Node id
n-e9e0cf63ac4aea4d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ac2697b1bcbae05c

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: 1,    4;
  2: 1,   20,     36;
  3: 1,   56,    441,      576;
  4: 1,  120,   2457,    13120,      14400;
  5: 1,  220,   9207,   122320,     526900,      518400;
  6: 1,  364,  27027,   711568,    7407400,    27832896,     25401600;
  7: 1,  560,  67158,  3065920,   61886825,   550499040,   1881701136,    1625702400;
  8: 1,  816, 147798, 10707008,  368478825,  6253988832,  49836284176,  158920556544,   131681894400;
  9: 1, 1140, 296514, 31989920, 1723584525, 49233358980, 739338552904, 5431396008960, 16423522073856, 13168189440000;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(4*n^4-8*n^3+4*n^2)*x-(9*n^6-36*n^5+54*n^4-36*n^3+9*n^2)*x^2)*P(n-2,x) + ((4*n^4-8*n^3+4*n^2)*x^2+(4*n^6-16*n^5+24*n^4-16*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2-2*n+1+(4*n^4-8*n^3+4*n^2)*x)*P(n-1,x) - (4*n^4-8*n^3+4*n^2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (4*n^4 - 8*n^3 + 4*n^2)*x - (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*P[n - 1, x] - (4*n^4 - 8*n^3 + 4*n^2)*x^2*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (4*n^4 - 8*n^3 + 4*n^2)*x - (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*v[i - 2] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*v[i - 1] - (4*n^4 - 8*n^3 + 4*n^2)*x^2*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.