Real-rooted atlas

A143685 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n x + 12\,n x^{2} + 2\,n x^{3} + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(160 + 372\,x + 268\,x^{2} + 60\,x^{3} + 4\,x^{4} - 160\,n - 228\,n x - 34\,n x^{2} + 41\,n x^{3} + 6\,n x^{4} - 120\,n^{2} x - 154\,n^{2} x^{2} - 34\,n^{2} x^{3} + 2\,n^{2} x^{4} - 20\,n^{3} x^{2} - 19\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(160 + 560\,x + 724\,x^{2} + 410\,x^{3} + 88\,x^{4} + 2\,x^{5} + 160\,n x + 440\,n x^{2} + 404\,n x^{3} + 127\,n x^{4} + 3\,n x^{5} + 40\,n^{2} x^{2} + 80\,n^{2} x^{3} + 41\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
21326342
344132724252
420582072014051890
5806403051002140230612312
634572074233795209804755342784078
714168991761990081674806188089284768551124
859449475592113095211946646119301465441502263293634794
924641022176906105240779268051582601776060299040234008101023619600
1010278611027861032136075480490803900471017669772434406054876034425101011153094374
1142734124700753216457604028234326027216276015383894450642844934860093042048460123212985173516
1217797573213570876827649288160219586017888491351214552016505393602126195336182440351484060658681476136315999138
137405585496272610240914409328823925924112062696108899631298449885906414350185842814861783297961022550448701217421440308736968
1430828986543160581101995902143547454660620675667922656149732493036681959895143456826003619292355574518530560726653294060112255203015256383081630
Technical details
Node id
n-4d2485358157dc4d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-077a68957f1c8b99

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,       2;
  2:     13,      26,       3;
  3:     44,     132,      72,       4;
  4:    205,     820,     720,     140,       5;
  5:    806,    4030,    5100,    2140,     230,       6;
  6:   3457,   20742,   33795,   20980,    4755,     342,      7;
  7:  14168,   99176,  199008,  167480,   61880,    8928,    476,     8;
  8:  59449,  475592, 1130952, 1194664,  611930,  146544,  15022,   632,   9;
  9: 246410, 2217690, 6105240, 7792680, 5158260, 1776060, 299040, 23400, 810, 10;
%F Formula
The row polynomials satisfy (12*n*x+(6*n^2+12*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(160*n-160+(120*n^2+228*n-372)*x+(20*n^3+154*n^2+34*n-268)*x^2+(19*n^3+34*n^2-41*n-60)*x^3-(2*n^2+6*n+4)*x^4)*P(n-1,x) + (160+(160*n+560)*x+(40*n^2+440*n+724)*x^2+(80*n^2+404*n+410)*x^3+(41*n^2+127*n+88)*x^4+(n^2+3*n+2)*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[(-(160*n - 160 + (120*n^2 + 228*n - 372)*x + (20*n^3 + 154*n^2 + 34*n - 268)*x^2 + (19*n^3 + 34*n^2 - 41*n - 60)*x^3 - (2*n^2 + 6*n + 4)*x^4)*P[n - 1, x] + (160 + (160*n + 560)*x + (40*n^2 + 440*n + 724)*x^2 + (80*n^2 + 404*n + 410)*x^3 + (41*n^2 + 127*n + 88)*x^4 + (n^2 + 3*n + 2)*x^5)*D[P[n - 1, x], x])/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*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] = (-(160*n - 160 + (120*n^2 + 228*n - 372)*x + (20*n^3 + 154*n^2 + 34*n - 268)*x^2 + (19*n^3 + 34*n^2 - 41*n - 60)*x^3 - (2*n^2 + 6*n + 4)*x^4)*v[i - 1] + (160 + (160*n + 560)*x + (40*n^2 + 440*n + 724)*x^2 + (80*n^2 + 404*n + 410)*x^3 + (41*n^2 + 127*n + 88)*x^4 + (n^2 + 3*n + 2)*x^5)*deriv(v[i - 1], x))/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*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.