Real-rooted atlas

A249250 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-48 - 96\,x - 64\,x^{2} - 16\,x^{3} + 12\,n x + 18\,n x^{2} + n x^{3} - n x^{4} + 12\,n^{2} x^{2} + 12\,n^{2} x^{3} + n^{2} x^{4} + 3\,n^{3} x^{3}\right)\,P_{n}(x) = \left(-48 - 96\,x - 64\,x^{2} - 16\,x^{3} - 144\,n - 420\,n x - 372\,n x^{2} - 95\,n x^{3} + 2\,n x^{4} - 108\,n^{2} x - 186\,n^{2} x^{2} - 78\,n^{2} x^{3} - 2\,n^{2} x^{4} - 18\,n^{3} x^{2} - 15\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(144 + 648\,x + 1092\,x^{2} + 846\,x^{3} + 288\,x^{4} + 31\,x^{5} + x^{6} + 144\,n x + 468\,n x^{2} + 528\,n x^{3} + 231\,n x^{4} + 28\,n x^{5} + n x^{6} + 36\,n^{2} x^{2} + 72\,n^{2} x^{3} + 39\,n^{2} x^{4} + 3\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
21024943
328967212208
47633637814415949
5208112816921008240184294
656836486840556821303602119135
715521152025920267841422038885042484412

All 15 rows

Technical details
Node id
n-819e9b831ad30df1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a62025445ba8b877

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,      6;
  2:    10,     24,      9;
  3:    28,     96,     72,     12;
  4:    76,    336,    378,    144,     15;
  5:   208,   1128,   1692,   1008,    240,     18;
  6:   568,   3648,   6840,   5568,   2130,    360,    21;
  7:  1552,  11520,  25920,  26784,  14220,   3888,   504,   24;
  8:  4240,  35712,  93744, 117504,  80100,  30672,  6426,  672,  27;
  9: 11584, 109152, 327456, 482688, 402480, 199368, 58716, 9888, 864, 30;
%F Formula
The row polynomials satisfy -(48-(12*n-96)*x-(12*n^2+18*n-64)*x^2-(3*n^3+12*n^2+n-16)*x^3-(n^2-n)*x^4)*P(n,x) = -(144*n+48+(108*n^2+420*n+96)*x+(18*n^3+186*n^2+372*n+64)*x^2+(15*n^3+78*n^2+95*n+16)*x^3+(2*n^2-2*n)*x^4)*P(n-1,x) + (144+(144*n+648)*x+(36*n^2+468*n+1092)*x^2+(72*n^2+528*n+846)*x^3+(39*n^2+231*n+288)*x^4+(3*n^2+28*n+31)*x^5+(n+1)*x^6)*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[(-(144*n + 48 + (108*n^2 + 420*n + 96)*x + (18*n^3 + 186*n^2 + 372*n + 64)*x^2 + (15*n^3 + 78*n^2 + 95*n + 16)*x^3 + (2*n^2 - 2*n)*x^4)*P[n - 1, x] + (144 + (144*n + 648)*x + (36*n^2 + 468*n + 1092)*x^2 + (72*n^2 + 528*n + 846)*x^3 + (39*n^2 + 231*n + 288)*x^4 + (3*n^2 + 28*n + 31)*x^5 + (n + 1)*x^6)*D[P[n - 1, x], x])/(-48 + (12*n - 96)*x + (12*n^2 + 18*n - 64)*x^2 + (3*n^3 + 12*n^2 + n - 16)*x^3 + (n^2 - n)*x^4)]];
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] = (-(144*n + 48 + (108*n^2 + 420*n + 96)*x + (18*n^3 + 186*n^2 + 372*n + 64)*x^2 + (15*n^3 + 78*n^2 + 95*n + 16)*x^3 + (2*n^2 - 2*n)*x^4)*v[i - 1] + (144 + (144*n + 648)*x + (36*n^2 + 468*n + 1092)*x^2 + (72*n^2 + 528*n + 846)*x^3 + (39*n^2 + 231*n + 288)*x^4 + (3*n^2 + 28*n + 31)*x^5 + (n + 1)*x^6)*deriv(v[i - 1], x))/(-48 + (12*n - 96)*x + (12*n^2 + 18*n - 64)*x^2 + (3*n^3 + 12*n^2 + n - 16)*x^3 + (n^2 - n)*x^4)); 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.