Real-rooted atlas

A154986: Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.

A154986 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 A154986
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(-n x + n^{2} x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
21416
311111124
412470241120
5145314314451720
61761079272810797615040
71119304516995169953045119140320
811767420804641867588046474201761362880
9124916164307124149086214908623071241616424913628800
10134032253991088903974619789944903974699108832253340139916800
11145159983280138143814474192824510192824510438144742801381599834511479001600
1215841053147118760177439471142988545629979216281429885456177439471711876010531458416227020800
1317411762541658142262157366784423828713450843064434508430644844238287162157366716581422176254741187178291200
141924283283359248241933769409413579402603031899665076146385975843031899665074135794026019337694093592482428328392411307674368000

Generated b-files: A154986, A154986 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(P_{n}(x) = \left(-n x + n^{2} x\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
Technical details
Node id
n-08ac2c6d9914754c · revision 2 · JSON
Definition source
oeis_catalog:A154986
derived:reversal (equal by e-b21254b1ce82559a)
Fingerprint
f6-454805add51456b0

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,   4,     1;
  3: 1,  11,    11,      1;
  4: 1,  24,    70,     24,       1;
  5: 1,  45,   314,    314,      45,       1;
  6: 1,  76,  1079,   2728,    1079,      76,      1;
  7: 1, 119,  3045,  16995,   16995,    3045,    119,     1;
  8: 1, 176,  7420,  80464,  186758,   80464,   7420,   176,   1;
  9: 1, 249, 16164, 307124, 1490862, 1490862, 307124, 16164, 249, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (n^2-n)*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 + x)*P[n - 1, x] + (n^2 - n)*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 + x)*v[i - 1] + (n^2 - n)*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.

A154986: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,     1;
  3: 1,  11,    11,      1;
  4: 1,  24,    70,     24,       1;
  5: 1,  45,   314,    314,      45,       1;
  6: 1,  76,  1079,   2728,    1079,      76,      1;
  7: 1, 119,  3045,  16995,   16995,    3045,    119,     1;
  8: 1, 176,  7420,  80464,  186758,   80464,   7420,   176,   1;
  9: 1, 249, 16164, 307124, 1490862, 1490862, 307124, 16164, 249, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (n^2-n)*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 + x)*P[n - 1, x] + (n^2 - n)*x*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (n^2 - n)*x*v[i - 2]); 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.

A154986: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   4,     1;
  3: 1,  11,    11,      1;
  4: 1,  24,    70,     24,       1;
  5: 1,  45,   314,    314,      45,       1;
  6: 1,  76,  1079,   2728,    1079,      76,      1;
  7: 1, 119,  3045,  16995,   16995,    3045,    119,     1;
  8: 1, 176,  7420,  80464,  186758,   80464,   7420,   176,   1;
  9: 1, 249, 16164, 307124, 1490862, 1490862, 307124, 16164, 249, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (n^2-n)*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 + x)*P[n - 1, x] + (n^2 - n)*x*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (n^2 - n)*x*v[i - 2]); 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-bad4de7d401fa939
  • 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-3d59ef840ca39edf, e-bad4de7d401fa939
  • 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-3d59ef840ca39edf, e-bad4de7d401fa939
  • 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-bad4de7d401fa939
  • 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-3d59ef840ca39edf

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 12 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