Real-rooted atlas

A144439 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4\,x^{2} - 8\,n x^{2}\right)\,P_{n-4}(x) + \left(-4\,x - 4\,x^{2} + 8\,n x + 32\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(4\,x^{2} - 4\,x^{3} - 8\,n x^{2} + 8\,n x^{3}\right)\,P_{n-3}'(x) + \left(-1 - 6\,x - 9\,x^{2} + 2\,n - 8\,n x - 6\,n x^{2} + 8\,n^{2} x + 20\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 8\,x^{2} - 6\,x^{3} + 4\,n x + 4\,n x^{2} - 8\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 6\,x - 16\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)
\(P_{2}(x) = 1 + 16\,x + 3\,x^{2}\)
\(P_{3}(x) = 1 + 62\,x + 93\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
2116320
3162934160
4120497840851596
51634721896241585619068
611928452611373447543557847265760
715814260019152378819047355200382034984234752
81174761419252146008243462859021901236331158869904975948880
9152466750691212749952051622811061947373222312416020018432236097101514139440
101157440389248711046419552675442225013711530720945619115020928391041167746137872001133218046912
11147236619926879382272523728072891565025743477248430034056789812916626504018511317837664229310259801312795275659776
12114171481011695814627427090169040054026354301113592424784084833702857348181232321627209724743156856772540370955131885028881320632325828800
1314251498510878777446805782935296467560937756594770912797817780119738458020320136843952098921563691967192935121875501287159030718204351510962763473714576592899610560
1411275455225707621489343577215270499264179637005947540452312152362941439542009361139629922416644666389936968691157923408718692021838119520141110307316458112111399693119892818641517268067536929792
Technical details
Node id
n-8e8b7bce030d60e7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-78732396ad9c62f4

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,     2;
  2: 1,    16,       3;
  3: 1,    62,      93,         4;
  4: 1,   204,     978,       408,         5;
  5: 1,   634,    7218,      9624,      1585,         6;
  6: 1,  1928,   45261,    137344,     75435,      5784,         7;
  7: 1,  5814,  260019,   1523788,   1904735,    520038,     20349,        8;
  8: 1, 17476, 1419252,  14600824,  34628590,  21901236,   3311588,    69904,      9;
  9: 1, 52466, 7506912, 127499520, 516228110, 619473732, 223124160, 20018432, 236097, 10;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n-4)*x^2*P(n-4,x) + ((8*n-4)*x-(16*n^2-32*n+4)*x^2)*P(n-3,x) - ((8*n-4)*x^2-(8*n-4)*x^3)*d/dx P(n-3,x) + (2*n-1+(8*n^2-8*n-6)*x-(8*n^3-20*n^2+6*n+9)*x^2)*P(n-2,x) + ((4*n-2)*x-(8*n^2-4*n-8)*x^2+(8*n^2-8*n-6)*x^3)*d/dx P(n-2,x) - (2-(8*n^2-16*n+6)*x)*P(n-1,x) + ((4*n-6)*x-(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 16*x + 3*x^2, P(3,x) = 1 + 62*x + 93*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 16*x + 3*x^2;
P[3, x_] := 1 + 62*x + 93*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 4)*x^2*P[n - 4, x] + ((8*n - 4)*x - (16*n^2 - 32*n + 4)*x^2)*P[n - 3, x] - ((8*n - 4)*x^2 - (8*n - 4)*x^3)*D[P[n - 3, x], x] + (2*n - 1 + (8*n^2 - 8*n - 6)*x - (8*n^3 - 20*n^2 + 6*n + 9)*x^2)*P[n - 2, x] + ((4*n - 2)*x - (8*n^2 - 4*n - 8)*x^2 + (8*n^2 - 8*n - 6)*x^3)*D[P[n - 2, x], x] - (2 - (8*n^2 - 16*n + 6)*x)*P[n - 1, x] + ((4*n - 6)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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 + 2*x; v[3] = 1 + 16*x + 3*x^2; v[4] = 1 + 62*x + 93*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 4)*x^2*v[i - 4] + ((8*n - 4)*x - (16*n^2 - 32*n + 4)*x^2)*v[i - 3] - ((8*n - 4)*x^2 - (8*n - 4)*x^3)*deriv(v[i - 3], x) + (2*n - 1 + (8*n^2 - 8*n - 6)*x - (8*n^3 - 20*n^2 + 6*n + 9)*x^2)*v[i - 2] + ((4*n - 2)*x - (8*n^2 - 4*n - 8)*x^2 + (8*n^2 - 8*n - 6)*x^3)*deriv(v[i - 2], x) - (2 - (8*n^2 - 16*n + 6)*x)*v[i - 1] + ((4*n - 6)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 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.