Real-rooted atlas

A257624: 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) = 3*x + 5.

A257624 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 A257624
\(P_{n}(x) = \left(5 + 2\,x + 3\,n x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15510
2258025130
31259159151252080
4625907020130907062539520
5312583185348410348410831853125869440
6156257279805246655975548052466557279801562521736000
77812561832157227280522576985522576985572272805618321578125608608000

Show all 15 stored rows

Generated b-files: A051607 (row sums), A257624, A257624 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-f3a4ae052af3da90 · revision 1 · JSON
Definition source
oeis_catalog:A257624
Fingerprint
f6-560c70a8d992c842

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:       5,         5;
  2:      25,        80,          25;
  3:     125,       915,         915,         125;
  4:     625,      9070,       20130,        9070,          625;
  5:    3125,     83185,      348410,      348410,        83185,         3125;
  6:   15625,    727980,     5246655,     9755480,      5246655,       727980,       15625;
  7:   78125,   6183215,    72272805,   225769855,    225769855,     72272805,     6183215,       78125;
  8:  390625,  51496970,   937214800,  4606234070,   7676175070,   4606234070,   937214800,    51496970,    390625;
  9: 1953125, 423303885, 11648284020, 86043217380, 222619657590, 222619657590, 86043217380, 11648284020, 423303885, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (5+(3*n+2)*x)*P(n-1,x) + (3*x-3*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[(5 + (3*n + 2)*x)*P[n - 1, x] + (3*x - 3*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] = (5 + (3*n + 2)*x)*v[i - 1] + (3*x - 3*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.

A257624: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,         5;
  2:      25,        80,          25;
  3:     125,       915,         915,         125;
  4:     625,      9070,       20130,        9070,          625;
  5:    3125,     83185,      348410,      348410,        83185,         3125;
  6:   15625,    727980,     5246655,     9755480,      5246655,       727980,       15625;
  7:   78125,   6183215,    72272805,   225769855,    225769855,     72272805,     6183215,       78125;
  8:  390625,  51496970,   937214800,  4606234070,   7676175070,   4606234070,   937214800,    51496970,    390625;
  9: 1953125, 423303885, 11648284020, 86043217380, 222619657590, 222619657590, 86043217380, 11648284020, 423303885, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (5+(3*n+2)*x)*P(n-1,x) + (3*x-3*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[(5 + (3*n + 2)*x)*P[n - 1, x] + (3*x - 3*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] = (5 + (3*n + 2)*x)*v[i - 1] + (3*x - 3*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.

A257624: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,         5;
  2:      25,        80,          25;
  3:     125,       915,         915,         125;
  4:     625,      9070,       20130,        9070,          625;
  5:    3125,     83185,      348410,      348410,        83185,         3125;
  6:   15625,    727980,     5246655,     9755480,      5246655,       727980,       15625;
  7:   78125,   6183215,    72272805,   225769855,    225769855,     72272805,     6183215,       78125;
  8:  390625,  51496970,   937214800,  4606234070,   7676175070,   4606234070,   937214800,    51496970,    390625;
  9: 1953125, 423303885, 11648284020, 86043217380, 222619657590, 222619657590, 86043217380, 11648284020, 423303885, 1953125;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+(3*n+2)*x)*P(n-1,x) + (3*x-3*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[(5 + (3*n + 2)*x)*P[n - 1, x] + (3*x - 3*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] = (5 + (3*n + 2)*x)*v[i - 1] + (3*x - 3*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-013c0ad711a68b34
  • 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-013c0ad711a68b34
  • 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-013c0ad711a68b34
  • 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-013c0ad711a68b34

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