Real-rooted atlas

A144438: Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A144438 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A144438
\(P_{n}(x) = \left(1 + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x) + \left(x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
21517
311414130
413389331157
5172413413721972
6115116323393163215116961
71310587422145221455874310156660
8162919943125456224843125456199436291516901
911268651716472991899096189909664729965171126815225670
1012547207554313050714152029230139951415202931305072075542547157999271
11151066494001445518596451500239047269239047269964515001445518564940051061701216922
121102252006913645222946154846722220047643336967576122200476436154846726452229420069131022519173819257
13120464614854528081461937371014851895609940641587158740415871587401895609940637371014852808146196148545204641129134686520
1414094318721892119904792921838990528151723095958463935053477668764215601463935053477151723095958218389905281199047929187218924094311946194117057

Generated b-files: A144438, A144438 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-2872b2f2cd7c5725 · revision 1 · JSON
Definition source
oeis_catalog:A144438
Fingerprint
f6-8e0b85d88ec2ca50

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,    5,     1;
  3: 1,   14,    14,      1;
  4: 1,   33,    89,     33,       1;
  5: 1,   72,   413,    413,      72,       1;
  6: 1,  151,  1632,   3393,    1632,     151,      1;
  7: 1,  310,  5874,  22145,   22145,    5874,    310,     1;
  8: 1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;
  9: 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + x*v[i - 2]); 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.

A144438: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    5,     1;
   4: 1,   14,    14,      1;
   5: 1,   33,    89,     33,       1;
   6: 1,   72,   413,    413,      72,       1;
   7: 1,  151,  1632,   3393,    1632,     151,      1;
   8: 1,  310,  5874,  22145,   22145,    5874,    310,     1;
   9: 1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;
  10: 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A144438: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    5,     1;
   4: 1,   14,    14,      1;
   5: 1,   33,    89,     33,       1;
   6: 1,   72,   413,    413,      72,       1;
   7: 1,  151,  1632,   3393,    1632,     151,      1;
   8: 1,  310,  5874,  22145,   22145,    5874,    310,     1;
   9: 1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;
  10: 1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x] + x*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x) + x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-d401dbe9093b268d
  • 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-d401dbe9093b268d
  • 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-d401dbe9093b268d
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-d401dbe9093b268d

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.

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