Real-rooted atlas

A299989: Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.

A299989 nonnegative coefficients only rational roots

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

Recurrence from OEIS A299989
\(P_{n}(x) = \left(3 + 4\,x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0011
103418
20924228164
302710817113657121512
408143297212008864001081614096
5024316204725792084305944281088017520132768
60729583220898442806169559472406361982468551640258241262144
7021872041286751222264383103469476421491281488140497521641418927443572812097152

Show all 15 stored rows

Generated b-files: A001018 (row sums), A299989 ?

Computed in your browser ?

Technical details
Node id
n-7e353d0275e3ebc5 · revision 1 · JSON
Definition source
oeis_catalog:A299989
Fingerprint
f6-45b1586921223be0

Observed on the stored rows ?

  • Degrees: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 (affine: slope 2, intercept 1)
  • 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 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 0,     1;
  1: 0,     3,      4,       1;
  2: 0,     9,     24,      22,       8,        1;
  3: 0,    27,    108,     171,     136,       57,       12,        1;
  4: 0,    81,    432,     972,    1200,      886,      400,      108,       16,        1;
  5: 0,   243,   1620,    4725,    7920,     8430,     5944,     2810,      880,      175,       20,       1;
  6: 0,   729,   5832,   20898,   44280,    61695,    59472,    40636,    19824,     6855,     1640,     258,      24,      1;
  7: 0,  2187,  20412,   86751,  222264,   383103,   469476,   421491,   281488,   140497,    52164,   14189,    2744,    357,     28,     1;
  8: 0,  6561,  69984,  344088, 1034208,  2125116,  3163104,  3525480,  2999904,  1968934,   999968,  391720,  117152,  26236,   4256,   472,   32,   1;
  9: 0, 19683, 236196, 1318761, 4548960, 10856268, 19023984, 25353972, 26264736, 21431898, 13875544, 7143966, 2918304, 939036, 234864, 44676, 6240, 603, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+4*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3 + 4*x + x^2)*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] = x; for(n = 1, N, my(i = n + 1); v[i] = (3 + 4*x + x^2)*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.

A299989: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 0,     1;
  1: 0,     3,      4,       1;
  2: 0,     9,     24,      22,       8,        1;
  3: 0,    27,    108,     171,     136,       57,       12,        1;
  4: 0,    81,    432,     972,    1200,      886,      400,      108,       16,        1;
  5: 0,   243,   1620,    4725,    7920,     8430,     5944,     2810,      880,      175,       20,       1;
  6: 0,   729,   5832,   20898,   44280,    61695,    59472,    40636,    19824,     6855,     1640,     258,      24,      1;
  7: 0,  2187,  20412,   86751,  222264,   383103,   469476,   421491,   281488,   140497,    52164,   14189,    2744,    357,     28,     1;
  8: 0,  6561,  69984,  344088, 1034208,  2125116,  3163104,  3525480,  2999904,  1968934,   999968,  391720,  117152,  26236,   4256,   472,   32,   1;
  9: 0, 19683, 236196, 1318761, 4548960, 10856268, 19023984, 25353972, 26264736, 21431898, 13875544, 7143966, 2918304, 939036, 234864, 44676, 6240, 603, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+4*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3 + 4*x + x^2)*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] = x; for(n = 1, N, my(i = n + 1); v[i] = (3 + 4*x + x^2)*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-0e7ef206328a472c
  • 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-0e7ef206328a472c
  • 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-0e7ef206328a472c

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

Used in 5 proof steps

No candidate equalities or related sequences.