Real-rooted atlas

A059473 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(36\,n x + 108\,n x^{2} + 54\,n x^{3} + 54\,n^{2} x^{2} + 81\,n^{2} x^{3} + 27\,n^{3} x^{3}\right)\,P_{n}(x) = \left(48 + 396\,x + 1152\,x^{2} + 1404\,x^{3} + 648\,x^{4} - 48\,n - 144\,n x + 306\,n x^{2} + 1296\,n x^{3} + 972\,n x^{4} - 108\,n^{2} x - 324\,n^{2} x^{2} + 324\,n^{2} x^{4} - 54\,n^{3} x^{2} - 108\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(16 + 168\,x + 672\,x^{2} + 1260\,x^{3} + 1080\,x^{4} + 324\,x^{5} + 48\,n x + 396\,n x^{2} + 1152\,n x^{3} + 1350\,n x^{4} + 486\,n x^{5} + 36\,n^{2} x^{2} + 216\,n^{2} x^{3} + 378\,n^{2} x^{4} + 162\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
218108108234
38072015128643176
435642721371616416648041240
515842376010281618921615552046656519552
670481268646909841721952217728013530243265926403744
731360658560432518413616640233539202230156811104128223948877630848

All 15 rows

Technical details
Node id
n-eab9d1a35e3e635f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ee51f20490618526

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:      4,       12;
  2:     18,      108,       108;
  3:     80,      720,      1512,       864;
  4:    356,     4272,     13716,     16416,       6480;
  5:   1584,    23760,    102816,    189216,     155520,      46656;
  6:   7048,   126864,    690984,   1721952,    2177280,    1353024,     326592;
  7:  31360,   658560,   4325184,  13616640,   23353920,   22301568,   11104128,    2239488;
  8: 139536,  3348864,  25751520,  98039808,  212809680,  275736960,  210651840,   87340032,  15116544;
  9: 620864, 16763328, 147723264, 660006144, 1735369920, 2847928896, 2951085312, 1874451456, 665127936, 100776960;
%F Formula
The row polynomials satisfy (36*n*x+(54*n^2+108*n)*x^2+(27*n^3+81*n^2+54*n)*x^3)*P(n,x) = -(48*n-48+(108*n^2+144*n-396)*x+(54*n^3+324*n^2-306*n-1152)*x^2+(108*n^3-1296*n-1404)*x^3-(324*n^2+972*n+648)*x^4)*P(n-1,x) + (16+(48*n+168)*x+(36*n^2+396*n+672)*x^2+(216*n^2+1152*n+1260)*x^3+(378*n^2+1350*n+1080)*x^4+(162*n^2+486*n+324)*x^5)*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[(-(48*n - 48 + (108*n^2 + 144*n - 396)*x + (54*n^3 + 324*n^2 - 306*n - 1152)*x^2 + (108*n^3 - 1296*n - 1404)*x^3 - (324*n^2 + 972*n + 648)*x^4)*P[n - 1, x] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 672)*x^2 + (216*n^2 + 1152*n + 1260)*x^3 + (378*n^2 + 1350*n + 1080)*x^4 + (162*n^2 + 486*n + 324)*x^5)*D[P[n - 1, x], x])/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*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] = (-(48*n - 48 + (108*n^2 + 144*n - 396)*x + (54*n^3 + 324*n^2 - 306*n - 1152)*x^2 + (108*n^3 - 1296*n - 1404)*x^3 - (324*n^2 + 972*n + 648)*x^4)*v[i - 1] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 672)*x^2 + (216*n^2 + 1152*n + 1260)*x^3 + (378*n^2 + 1350*n + 1080)*x^4 + (162*n^2 + 486*n + 324)*x^5)*deriv(v[i - 1], x))/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.