Real-rooted atlas

A257623: Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.

A257623 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 A257623
\(P_{n}(x) = \left(3 - 2\,x + 5\,n x\right)\,P_{n-1}(x) + \left(5\,x - 5\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2948966
327501501271056
48144941302644948122176
52433781525023025023037815243576576
672930932441227359008280412273530932472917873856
721872498649622566272569719452569719456225662724986492187643458816

Show all 15 stored rows

Generated b-files: A257623, A257623 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-4155b181fd90d601 · revision 1 · JSON
Definition source
oeis_catalog:A257623
Fingerprint
f6-90d57b3ffa6012d6

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:     3,         3;
  2:     9,        48,           9;
  3:    27,       501,         501,           27;
  4:    81,      4494,       13026,         4494,           81;
  5:   243,     37815,      250230,       250230,        37815,          243;
  6:   729,    309324,     4122735,      9008280,      4122735,       309324,          729;
  7:  2187,   2498649,    62256627,    256971945,    256971945,     62256627,      2498649,        2187;
  8:  6561,  20072298,   891791568,   6368680566,  11820709470,   6368680566,    891791568,    20072298,      6561;
  9: 19683, 160860507, 12356037708, 144065371932, 450199373658, 450199373658, 144065371932, 12356037708, 160860507, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3+(5*n-2)*x)*P(n-1,x) + (5*x-5*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[(3 + (5*n - 2)*x)*P[n - 1, x] + (5*x - 5*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] = (3 + (5*n - 2)*x)*v[i - 1] + (5*x - 5*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.

A257623: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,         3;
  2:     9,        48,           9;
  3:    27,       501,         501,           27;
  4:    81,      4494,       13026,         4494,           81;
  5:   243,     37815,      250230,       250230,        37815,          243;
  6:   729,    309324,     4122735,      9008280,      4122735,       309324,          729;
  7:  2187,   2498649,    62256627,    256971945,    256971945,     62256627,      2498649,        2187;
  8:  6561,  20072298,   891791568,   6368680566,  11820709470,   6368680566,    891791568,    20072298,      6561;
  9: 19683, 160860507, 12356037708, 144065371932, 450199373658, 450199373658, 144065371932, 12356037708, 160860507, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3+(5*n-2)*x)*P(n-1,x) + (5*x-5*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[(3 + (5*n - 2)*x)*P[n - 1, x] + (5*x - 5*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] = (3 + (5*n - 2)*x)*v[i - 1] + (5*x - 5*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.

A257623: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,         3;
  2:     9,        48,           9;
  3:    27,       501,         501,           27;
  4:    81,      4494,       13026,         4494,           81;
  5:   243,     37815,      250230,       250230,        37815,          243;
  6:   729,    309324,     4122735,      9008280,      4122735,       309324,          729;
  7:  2187,   2498649,    62256627,    256971945,    256971945,     62256627,      2498649,        2187;
  8:  6561,  20072298,   891791568,   6368680566,  11820709470,   6368680566,    891791568,    20072298,      6561;
  9: 19683, 160860507, 12356037708, 144065371932, 450199373658, 450199373658, 144065371932, 12356037708, 160860507, 19683;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+(5*n-2)*x)*P(n-1,x) + (5*x-5*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[(3 + (5*n - 2)*x)*P[n - 1, x] + (5*x - 5*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] = (3 + (5*n - 2)*x)*v[i - 1] + (5*x - 5*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-d9277c75939e8d2f
  • 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-d9277c75939e8d2f
  • 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-d9277c75939e8d2f
  • 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-d9277c75939e8d2f

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