Real-rooted atlas

A081580 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n x + 12\,n x^{2} + 2\,n x^{3} + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(96 + 228\,x + 172\,x^{2} + 44\,x^{3} + 4\,x^{4} - 96\,n - 132\,n x - 10\,n x^{2} + 33\,n x^{3} + 6\,n x^{4} - 72\,n^{2} x - 90\,n^{2} x^{2} - 18\,n^{2} x^{3} + 2\,n^{2} x^{4} - 12\,n^{3} x^{2} - 11\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(96 + 336\,x + 436\,x^{2} + 250\,x^{3} + 56\,x^{4} + 2\,x^{5} + 96\,n x + 264\,n x^{2} + 244\,n x^{3} + 79\,n x^{4} + 3\,n x^{5} + 24\,n^{2} x^{2} + 48\,n^{2} x^{3} + 25\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2918330
32884484164
4101404372925974
534217102220102015065448
6118971341189579402175222730562
740882861658416516402140039843088168460

All 15 rows

Technical details
Node id
n-94e283a81cf0db74 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-435b9980d59625b6

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:     9,     18,       3;
  3:    28,     84,      48,       4;
  4:   101,    404,     372,      92,       5;
  5:   342,   1710,    2220,    1020,     150,      6;
  6:  1189,   7134,   11895,    7940,    2175,    222,     7;
  7:  4088,  28616,   58416,   51640,   21400,   3984,   308,     8;
  8: 14121, 112968,  272736,  300168,  167850,  47808,  6594,   408,   9;
  9: 48682, 438138, 1221624, 1611624, 1140660, 445356, 93744, 10152, 522, 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) = -(96*n-96+(72*n^2+132*n-228)*x+(12*n^3+90*n^2+10*n-172)*x^2+(11*n^3+18*n^2-33*n-44)*x^3-(2*n^2+6*n+4)*x^4)*P(n-1,x) + (96+(96*n+336)*x+(24*n^2+264*n+436)*x^2+(48*n^2+244*n+250)*x^3+(25*n^2+79*n+56)*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[(-(96*n - 96 + (72*n^2 + 132*n - 228)*x + (12*n^3 + 90*n^2 + 10*n - 172)*x^2 + (11*n^3 + 18*n^2 - 33*n - 44)*x^3 - (2*n^2 + 6*n + 4)*x^4)*P[n - 1, x] + (96 + (96*n + 336)*x + (24*n^2 + 264*n + 436)*x^2 + (48*n^2 + 244*n + 250)*x^3 + (25*n^2 + 79*n + 56)*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] = (-(96*n - 96 + (72*n^2 + 132*n - 228)*x + (12*n^3 + 90*n^2 + 10*n - 172)*x^2 + (11*n^3 + 18*n^2 - 33*n - 44)*x^3 - (2*n^2 + 6*n + 4)*x^4)*v[i - 1] + (96 + (96*n + 336)*x + (24*n^2 + 264*n + 436)*x^2 + (48*n^2 + 244*n + 250)*x^3 + (25*n^2 + 79*n + 56)*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.