Real-rooted atlas

A144438 → D+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(\left(8\right)\,P_{n}(x) = \left(-4 - 14\,x - 14\,x^{2} - 10\,n - 5\,n x + 8\,n x^{3} + 6\,n^{2} + 3\,n^{2} x - 3\,n^{2} x^{3}\right)\,P_{n-4}(x) + \left(-2\,x - 6\,x^{2} - 5\,n x + 8\,n x^{3} + 3\,n^{2} x - 3\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(10 + 38\,x + 48\,x^{2} - 16\,x^{3} + 21\,n - 26\,n x - 14\,n x^{2} - 16\,n x^{3} - 25\,n^{2} - 5\,n^{2} x - 3\,n^{2} x^{2} + 14\,n^{2} x^{3} + 6\,n^{3} + 3\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-6 - 8\,x + 32\,x^{2} + 6\,x^{3} - 15\,n + 23\,n x + 7\,n x^{2} - 24\,n x^{3} - 8\,n x^{4} + 9\,n^{2} - 20\,n^{2} x - 6\,n^{2} x^{2} + 17\,n^{2} x^{3} + 3\,n^{2} x^{4} + 3\,n^{3} x - 3\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(-2\,x - 4\,x^{2} + 6\,x^{3} - 5\,n x + 5\,n x^{2} + 8\,n x^{3} - 8\,n x^{4} + 3\,n^{2} x - 3\,n^{2} x^{2} - 3\,n^{2} x^{3} + 3\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(-6 + 42\,x - 36\,x^{2} + 16\,x^{3} - 7\,n + 38\,n x + 20\,n x^{2} - 16\,n x^{3} - 3\,n^{2} - 12\,n^{2} x - 5\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2 + 26\,x - 32\,x^{2} - 4\,x^{3} + 16\,x^{4} + 5\,n - 6\,n x - 2\,n x^{2} - 3\,n^{2} + 3\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(8\,x^{2} - 16\,x^{3} + 8\,x^{4}\right)\,P_{n-2}''(x) + \left(20 - 44\,x + 16\,x^{2} + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 6 + 7\,x + x^{2}\)
\(P_{3}(x) = 15 + 42\,x + 17\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
267114
3154217175
434211188371471
57389816527017713402
615234151181199212387157127844
731112058723091107255151577343171254970
86304051539631110248288521232451142434663712584505
91269131610200706882436831139457657828901103496753151277128741185
10254841765595990755973862312922200410792616936065578479093923047725571347995626
1151071303906440149554002611851291687845167333088391420776921209298020299785700460511714557909993
1210226402405119557379525264609821171572288722438102209189100092627143925019119618531884591424211938810237164216734799
13204651231755484859240215229220559985175985152684790518463326972699201932359539885259001277165452476753484486146394113204771968510148900
14409443748472736158656798855501004178045447031829353334168205145284562684438024464341718294429170993701130012383502851774714237106331925415140957115569552936456

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Technical details
Node id
n-b8ad7ea132013cd3 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e2e85b9005326116

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    2,      1;
  2:    6,      7,       1;
  3:   15,     42,      17,       1;
  4:   34,    211,     188,      37,        1;
  5:   73,    898,    1652,     701,       77,       1;
  6:  152,   3415,   11811,    9921,     2387,     157,       1;
  7:  311,  12058,   72309,  110725,    51515,    7734,     317,     1;
  8:  630,  40515,  396311, 1024828,   852123,  245114,   24346,   637,    1;
  9: 1269, 131610, 2007068, 8243683, 11394576, 5782890, 1103496, 75315, 1277, 1;
%F Formula
The row polynomials satisfy 8*P(n,x) = (6*n^2-10*n-4+(3*n^2-5*n-14)*x-14*x^2-(3*n^2-8*n)*x^3)*P(n-4,x) + ((3*n^2-5*n-2)*x-6*x^2-(3*n^2-8*n)*x^3)*d/dx P(n-4,x) + (6*n^3-25*n^2+21*n+10+(3*n^3-5*n^2-26*n+38)*x-(3*n^2+14*n-48)*x^2-(3*n^3-14*n^2+16*n+16)*x^3)*P(n-3,x) + (9*n^2-15*n-6+(3*n^3-20*n^2+23*n-8)*x-(6*n^2-7*n-32)*x^2-(3*n^3-17*n^2+24*n-6)*x^3+(3*n^2-8*n)*x^4)*d/dx P(n-3,x) + ((3*n^2-5*n-2)*x-(3*n^2-5*n+4)*x^2-(3*n^2-8*n-6)*x^3+(3*n^2-8*n)*x^4)*d^2/dx^2 P(n-3,x) - (3*n^2+7*n+6+(12*n^2-38*n-42)*x+(5*n^2-20*n+36)*x^2+(16*n-16)*x^3)*P(n-2,x) - (3*n^2-5*n-2+(6*n-26)*x-(3*n^2-2*n-32)*x^2+4*x^3-16*x^4)*d/dx P(n-2,x) + (8*x^2-16*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (6*n+20+(16*n-44)*x+16*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 6 + 7*x + x^2, P(3,x) = 15 + 42*x + 17*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 6 + 7*x + x^2;
P[3, x_] := 15 + 42*x + 17*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 10*n - 4 + (3*n^2 - 5*n - 14)*x - 14*x^2 - (3*n^2 - 8*n)*x^3)*P[n - 4, x] + ((3*n^2 - 5*n - 2)*x - 6*x^2 - (3*n^2 - 8*n)*x^3)*D[P[n - 4, x], x] + (6*n^3 - 25*n^2 + 21*n + 10 + (3*n^3 - 5*n^2 - 26*n + 38)*x - (3*n^2 + 14*n - 48)*x^2 - (3*n^3 - 14*n^2 + 16*n + 16)*x^3)*P[n - 3, x] + (9*n^2 - 15*n - 6 + (3*n^3 - 20*n^2 + 23*n - 8)*x - (6*n^2 - 7*n - 32)*x^2 - (3*n^3 - 17*n^2 + 24*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*D[P[n - 3, x], x] + ((3*n^2 - 5*n - 2)*x - (3*n^2 - 5*n + 4)*x^2 - (3*n^2 - 8*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*D[P[n - 3, x], {x, 2}] - (3*n^2 + 7*n + 6 + (12*n^2 - 38*n - 42)*x + (5*n^2 - 20*n + 36)*x^2 + (16*n - 16)*x^3)*P[n - 2, x] - (3*n^2 - 5*n - 2 + (6*n - 26)*x - (3*n^2 - 2*n - 32)*x^2 + 4*x^3 - 16*x^4)*D[P[n - 2, x], x] + (8*x^2 - 16*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (6*n + 20 + (16*n - 44)*x + 16*x^2)*P[n - 1, x])/(8)]];
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] = 2 + x; v[3] = 6 + 7*x + x^2; v[4] = 15 + 42*x + 17*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((6*n^2 - 10*n - 4 + (3*n^2 - 5*n - 14)*x - 14*x^2 - (3*n^2 - 8*n)*x^3)*v[i - 4] + ((3*n^2 - 5*n - 2)*x - 6*x^2 - (3*n^2 - 8*n)*x^3)*deriv(v[i - 4], x) + (6*n^3 - 25*n^2 + 21*n + 10 + (3*n^3 - 5*n^2 - 26*n + 38)*x - (3*n^2 + 14*n - 48)*x^2 - (3*n^3 - 14*n^2 + 16*n + 16)*x^3)*v[i - 3] + (9*n^2 - 15*n - 6 + (3*n^3 - 20*n^2 + 23*n - 8)*x - (6*n^2 - 7*n - 32)*x^2 - (3*n^3 - 17*n^2 + 24*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*deriv(v[i - 3], x) + ((3*n^2 - 5*n - 2)*x - (3*n^2 - 5*n + 4)*x^2 - (3*n^2 - 8*n - 6)*x^3 + (3*n^2 - 8*n)*x^4)*deriv(deriv(v[i - 3], x), x) - (3*n^2 + 7*n + 6 + (12*n^2 - 38*n - 42)*x + (5*n^2 - 20*n + 36)*x^2 + (16*n - 16)*x^3)*v[i - 2] - (3*n^2 - 5*n - 2 + (6*n - 26)*x - (3*n^2 - 2*n - 32)*x^2 + 4*x^3 - 16*x^4)*deriv(v[i - 2], x) + (8*x^2 - 16*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n + 20 + (16*n - 44)*x + 16*x^2)*v[i - 1])/(8)); 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-aa88e1a16f7cc7e2
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-aa88e1a16f7cc7e2
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-aa88e1a16f7cc7e2
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-aa88e1a16f7cc7e2

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.