Real-rooted atlas

A257608: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.

A257608 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A257608
\(P_{n}(x) = \left(1 - 8\,x + 9\,n x\right)\,P_{n-1}(x) + \left(9\,x - 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
2120122
312192191440
41221883222218112760
5122217220222220222222171484880
612222165006247123324325006247222216122789360
712222215105340629530539235530539235105340629222221511276204160

Show all 15 stored rows

Generated b-files: A084949 (row sums), A257608, A257608 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-dcd2d11144e6ea21 · revision 1 · JSON
Definition source
oeis_catalog:A257608
Fingerprint
f6-bb7d69d9a9c0913d

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: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

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: 1;
  1: 1,         1;
  2: 1,        20,           1;
  3: 1,       219,         219,            1;
  4: 1,      2218,        8322,         2218,             1;
  5: 1,     22217,      220222,       220222,         22217,             1;
  6: 1,    222216,     5006247,     12332432,       5006247,        222216,            1;
  7: 1,   2222215,   105340629,    530539235,     530539235,     105340629,      2222215,           1;
  8: 1,  22222214,  2123693776,  19700767514,   39259903390,   19700767514,   2123693776,    22222214,         1;
  9: 1, 222222213, 41772403440, 668424648072, 2358851731074, 2358851731074, 668424648072, 41772403440, 222222213, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(9*n-8)*x)*P(n-1,x) + (9*x-9*x^2)*d/dx 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[(1 + (9*n - 8)*x)*P[n - 1, x] + (9*x - 9*x^2)*D[P[n - 1, x], x]];
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] = (1 + (9*n - 8)*x)*v[i - 1] + (9*x - 9*x^2)*deriv(v[i - 1], x)); 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.

A257608: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,         1;
  2: 1,        20,           1;
  3: 1,       219,         219,            1;
  4: 1,      2218,        8322,         2218,             1;
  5: 1,     22217,      220222,       220222,         22217,             1;
  6: 1,    222216,     5006247,     12332432,       5006247,        222216,            1;
  7: 1,   2222215,   105340629,    530539235,     530539235,     105340629,      2222215,           1;
  8: 1,  22222214,  2123693776,  19700767514,   39259903390,   19700767514,   2123693776,    22222214,         1;
  9: 1, 222222213, 41772403440, 668424648072, 2358851731074, 2358851731074, 668424648072, 41772403440, 222222213, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(9*n-8)*x)*P(n-1,x) + (9*x-9*x^2)*d/dx 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[(1 + (9*n - 8)*x)*P[n - 1, x] + (9*x - 9*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (1 + (9*n - 8)*x)*v[i - 1] + (9*x - 9*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A257608: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,         1;
  2: 1,        20,           1;
  3: 1,       219,         219,            1;
  4: 1,      2218,        8322,         2218,             1;
  5: 1,     22217,      220222,       220222,         22217,             1;
  6: 1,    222216,     5006247,     12332432,       5006247,        222216,            1;
  7: 1,   2222215,   105340629,    530539235,     530539235,     105340629,      2222215,           1;
  8: 1,  22222214,  2123693776,  19700767514,   39259903390,   19700767514,   2123693776,    22222214,         1;
  9: 1, 222222213, 41772403440, 668424648072, 2358851731074, 2358851731074, 668424648072, 41772403440, 222222213, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(9*n-8)*x)*P(n-1,x) + (9*x-9*x^2)*d/dx 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[(1 + (9*n - 8)*x)*P[n - 1, x] + (9*x - 9*x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (1 + (9*n - 8)*x)*v[i - 1] + (9*x - 9*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-e4e9153fc21b93ec
  • 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-e4e9153fc21b93ec
  • 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-e4e9153fc21b93ec
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-e4e9153fc21b93ec

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 3 proof steps

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