Real-rooted atlas

A122431 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(1 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 8 + 48\,x + 96\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10Pn(1)
08489664216
18802884482561080
281126081600204810245400
381441056403284489216409627000
481761632825624576430084096016384135000
582082336147845760014131221299218022465536675000
6824031682412811673637171277824010321927864322621443375000
782724128368002132488386562265088414515249152003407872104857616875000

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-c77d53cf93e5da02 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-751aae81c1f2029f

Observed on the stored rows ?

  • Degrees: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 (affine: slope 1, intercept 3)
  • 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): yes

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: 8,  48,   96,    64;
  1: 8,  80,  288,   448,    256;
  2: 8, 112,  608,  1600,   2048,    1024;
  3: 8, 144, 1056,  4032,   8448,    9216,     4096;
  4: 8, 176, 1632,  8256,  24576,   43008,    40960,    16384;
  5: 8, 208, 2336, 14784,  57600,  141312,   212992,   180224,    65536;
  6: 8, 240, 3168, 24128, 116736,  371712,   778240,  1032192,   786432,    262144;
  7: 8, 272, 4128, 36800, 213248,  838656,  2265088,  4145152,  4915200,   3407872,   1048576;
  8: 8, 304, 5216, 53312, 360448, 1691648,  5619712, 13205504, 21495808,  23068672,  14680064,  4194304;
  9: 8, 336, 6432, 74176, 573696, 3133440, 12386304, 35684352, 74317824, 109051904, 106954752, 62914560, 16777216;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 48*x + 96*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 8 + 48*x + 96*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8 + 48*x + 96*x^2 + 64*x^3; for(n = 1, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 1]); 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-4464410b9667d6e5
  • 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-4464410b9667d6e5
  • 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-4464410b9667d6e5

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.