Real-rooted atlas

A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

A178822 palindromic γ-positive nonnegative coefficients only rational roots

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

Recurrence from OEIS A178822
\(\left(n\right)\,P_{n}(x) = \left(5 + 5\,x + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16612
221422184
35616816856448
41265047565041262016
525212602520252012602528064
64622772693092406930277246229568
77925544166322772027720166325544792101376
81287102963603672072900907207236036102961287329472
920021801872072168168252252252252168168720721801820021025024
103003300301351353603606306307567566306303603601351353003030033075072
1143684804824024072072014414402018016201801614414407207202402404804843688945664
12618874256408408136136030630604900896571771249008963063060136136040840874256618825346048
138568111384668304245044861261201102701614702688147026881102701661261202450448668304111384856870189056
141162816279210581484232592116396282327925634918884399072963491888423279256116396284232592105814816279211628190513152

Generated b-files: A054849 (row sums), A178822, A178822 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-97228a88c81561dc · revision 2 · JSON
Definition source
oeis_catalog:A178822
Fingerprint
f6-77a7ed3b1d5f257e

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): yes

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:    6,     6;
  2:   21,    42,    21;
  3:   56,   168,   168,     56;
  4:  126,   504,   756,    504,    126;
  5:  252,  1260,  2520,   2520,   1260,    252;
  6:  462,  2772,  6930,   9240,   6930,   2772,    462;
  7:  792,  5544, 16632,  27720,  27720,  16632,   5544,   792;
  8: 1287, 10296, 36036,  72072,  90090,  72072,  36036, 10296,  1287;
  9: 2002, 18018, 72072, 168168, 252252, 252252, 168168, 72072, 18018, 2002;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+5+(n+5)*x)*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[Together[((n + 5 + (n + 5)*x)*P[n - 1, x])/(n)]];
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] = ((n + 5 + (n + 5)*x)*v[i - 1])/(n)); 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.

A178822: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    1;
  1:    6,     6;
  2:   21,    42,    21;
  3:   56,   168,   168,     56;
  4:  126,   504,   756,    504,    126;
  5:  252,  1260,  2520,   2520,   1260,    252;
  6:  462,  2772,  6930,   9240,   6930,   2772,    462;
  7:  792,  5544, 16632,  27720,  27720,  16632,   5544,   792;
  8: 1287, 10296, 36036,  72072,  90090,  72072,  36036, 10296,  1287;
  9: 2002, 18018, 72072, 168168, 252252, 252252, 168168, 72072, 18018, 2002;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+5+(n+5)*x)*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[Together[((n + 5 + (n + 5)*x)*P[n - 1, x])/(n)]];
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] = ((n + 5 + (n + 5)*x)*v[i - 1])/(n)); 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.

A178822: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    1;
  1:    6,     6;
  2:   21,    42,    21;
  3:   56,   168,   168,     56;
  4:  126,   504,   756,    504,    126;
  5:  252,  1260,  2520,   2520,   1260,    252;
  6:  462,  2772,  6930,   9240,   6930,   2772,    462;
  7:  792,  5544, 16632,  27720,  27720,  16632,   5544,   792;
  8: 1287, 10296, 36036,  72072,  90090,  72072,  36036, 10296,  1287;
  9: 2002, 18018, 72072, 168168, 252252, 252252, 168168, 72072, 18018, 2002;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (n+5+(n+5)*x)*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[Together[((n + 5 + (n + 5)*x)*P[n - 1, x])/(n)]];
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] = ((n + 5 + (n + 5)*x)*v[i - 1])/(n)); 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-548a05679c99d179
  • 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-0d5dad6bd99a01a4, e-548a05679c99d179
  • 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-0d5dad6bd99a01a4, e-548a05679c99d179
  • 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-548a05679c99d179
  • 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-0d5dad6bd99a01a4

Proof chains

2 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 17 proof steps

Same fingerprint (59)

Equal up to content, sign, reversal and a small index shift.

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