Real-rooted atlas

A152842 → D+1

nonnegative coefficients

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

Recurrence
\(\left(-11 - 58\,x - 36\,x^{2} + 36\,x^{3} - 11\,n + 18\,n x + 54\,n x^{2} + 12\,n^{2} + 18\,n^{2} x\right)\,P_{n}(x) = \left(-26 - 18\,x + 106\,x^{2} - 282\,x^{3} - 576\,x^{4} - 216\,x^{5} - 106\,n - 550\,n x - 931\,n x^{2} - 132\,n x^{3} + 522\,n x^{4} + 216\,n x^{5} + 108\,n^{2} + 498\,n^{2} x + 867\,n^{2} x^{2} + 612\,n^{2} x^{3} + 162\,n^{2} x^{4} - 16\,n^{3} - 48\,n^{3} x - 36\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-15 - 122\,x - 543\,x^{2} - 1546\,x^{3} - 2262\,x^{4} - 1368\,x^{5} - 216\,x^{6} - 56\,n - 456\,n x - 1324\,n x^{2} - 1608\,n x^{3} - 684\,n x^{4} + 16\,n^{2} + 112\,n^{2} x + 276\,n^{2} x^{2} + 288\,n^{2} x^{3} + 108\,n^{2} x^{4}\right)\,P_{n-1}'(x) + \left(-1 - 12\,x - 72\,x^{2} - 256\,x^{3} - 501\,x^{4} - 468\,x^{5} - 162\,x^{6} - 4\,n - 44\,n x - 184\,n x^{2} - 360\,n x^{3} - 324\,n x^{4} - 108\,n x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
2510318
383742996
495294609224
512103316414216271088
613126465820711270272432
71620110142623368427279188111264

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-7fa0a8eeaf34c691 · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2a74beb062736e56

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:  4,   3;
  2:  5,  10,    3;
  3:  8,  37,   42,    9;
  4:  9,  52,   94,   60,     9;
  5: 12, 103,  316,  414,   216,    27;
  6: 13, 126,  465,  820,   711,   270,    27;
  7: 16, 201, 1014, 2623,  3684,  2727,   918,    81;
  8: 17, 232, 1308, 3944,  6886,  7032,  3996,  1080,   81;
  9: 20, 331, 2328, 9068, 21376, 31290, 28008, 14364, 3564, 243;
%F Formula
The row polynomials satisfy (12*n^2-11*n-11+(18*n^2+18*n-58)*x+(54*n-36)*x^2+36*x^3)*P(n,x) = -(16*n^3-108*n^2+106*n+26+(48*n^3-498*n^2+550*n+18)*x+(36*n^3-867*n^2+931*n-106)*x^2-(612*n^2-132*n-282)*x^3-(162*n^2+522*n-576)*x^4-(216*n-216)*x^5)*P(n-1,x) + (16*n^2-56*n-15+(112*n^2-456*n-122)*x+(276*n^2-1324*n-543)*x^2+(288*n^2-1608*n-1546)*x^3+(108*n^2-684*n-2262)*x^4-1368*x^5-216*x^6)*d/dx P(n-1,x) - (4*n+1+(44*n+12)*x+(184*n+72)*x^2+(360*n+256)*x^3+(324*n+501)*x^4+(108*n+468)*x^5+162*x^6)*d^2/dx^2 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[(-(16*n^3 - 108*n^2 + 106*n + 26 + (48*n^3 - 498*n^2 + 550*n + 18)*x + (36*n^3 - 867*n^2 + 931*n - 106)*x^2 - (612*n^2 - 132*n - 282)*x^3 - (162*n^2 + 522*n - 576)*x^4 - (216*n - 216)*x^5)*P[n - 1, x] + (16*n^2 - 56*n - 15 + (112*n^2 - 456*n - 122)*x + (276*n^2 - 1324*n - 543)*x^2 + (288*n^2 - 1608*n - 1546)*x^3 + (108*n^2 - 684*n - 2262)*x^4 - 1368*x^5 - 216*x^6)*D[P[n - 1, x], x] - (4*n + 1 + (44*n + 12)*x + (184*n + 72)*x^2 + (360*n + 256)*x^3 + (324*n + 501)*x^4 + (108*n + 468)*x^5 + 162*x^6)*D[P[n - 1, x], {x, 2}])/(12*n^2 - 11*n - 11 + (18*n^2 + 18*n - 58)*x + (54*n - 36)*x^2 + 36*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] = (-(16*n^3 - 108*n^2 + 106*n + 26 + (48*n^3 - 498*n^2 + 550*n + 18)*x + (36*n^3 - 867*n^2 + 931*n - 106)*x^2 - (612*n^2 - 132*n - 282)*x^3 - (162*n^2 + 522*n - 576)*x^4 - (216*n - 216)*x^5)*v[i - 1] + (16*n^2 - 56*n - 15 + (112*n^2 - 456*n - 122)*x + (276*n^2 - 1324*n - 543)*x^2 + (288*n^2 - 1608*n - 1546)*x^3 + (108*n^2 - 684*n - 2262)*x^4 - 1368*x^5 - 216*x^6)*deriv(v[i - 1], x) - (4*n + 1 + (44*n + 12)*x + (184*n + 72)*x^2 + (360*n + 256)*x^3 + (324*n + 501)*x^4 + (108*n + 468)*x^5 + 162*x^6)*deriv(deriv(v[i - 1], x), x))/(12*n^2 - 11*n - 11 + (18*n^2 + 18*n - 58)*x + (54*n - 36)*x^2 + 36*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)

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-c18e1ce17600bec1
  • 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-c18e1ce17600bec1
  • 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-c18e1ce17600bec1

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