Real-rooted atlas

A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

A196347 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A196347
\(P_{n}(x) = \left(n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
22428
361818648
424961449624384
5120600120012006001203840
67204320108001440010800432072046080
7504035280105840176400176400105840352805040645120
840320322560112896022579202822400225792011289603225604032010321920
936288032659201306368030481920457228804572288030481920130636803265920362880185794560
103628800362880001632960004354560007620480009144576007620480004354560001632960003628800036288003715891200
11399168004390848002195424000658627200013172544000184415616001844156160013172544000658627200021954240004390848003991680081749606400
124790016005748019200316141056001053803520002371057920003793692672004425974784003793692672002371057920001053803520003161410560057480192004790016001961990553600
13622702080080951270400485707622400178092794880044523198720008014175769600106855676928001068556769280080141757696004452319872000178092794880048570762240080951270400622702080051011754393600
1487178291200122049607680079332244992003173289799680087265469491200174530938982400261796408473600299195895398400261796408473600174530938982400872654694912003173289799680079332244992001220496076800871782912001428329123020800

Generated b-files: A000165 (row sums), A196347, A196347 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-cb5d49e71e46f2ed · revision 3 · JSON
Definition source
oeis_catalog:A196347
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:      1,       1;
  2:      2,       4,        2;
  3:      6,      18,       18,        6;
  4:     24,      96,      144,       96,       24;
  5:    120,     600,     1200,     1200,      600,      120;
  6:    720,    4320,    10800,    14400,    10800,     4320,      720;
  7:   5040,   35280,   105840,   176400,   176400,   105840,    35280,     5040;
  8:  40320,  322560,  1128960,  2257920,  2822400,  2257920,  1128960,   322560,   40320;
  9: 362880, 3265920, 13063680, 30481920, 45722880, 45722880, 30481920, 13063680, 3265920, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (n+n*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[(n + n*x)*P[n - 1, 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] = (n + n*x)*v[i - 1]); 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.

A196347: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      2,       4,        2;
  3:      6,      18,       18,        6;
  4:     24,      96,      144,       96,       24;
  5:    120,     600,     1200,     1200,      600,      120;
  6:    720,    4320,    10800,    14400,    10800,     4320,      720;
  7:   5040,   35280,   105840,   176400,   176400,   105840,    35280,     5040;
  8:  40320,  322560,  1128960,  2257920,  2822400,  2257920,  1128960,   322560,   40320;
  9: 362880, 3265920, 13063680, 30481920, 45722880, 45722880, 30481920, 13063680, 3265920, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (n+n*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[(n + n*x)*P[n - 1, 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] = (n + n*x)*v[i - 1]); 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.

A196347: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      2,       4,        2;
  3:      6,      18,       18,        6;
  4:     24,      96,      144,       96,       24;
  5:    120,     600,     1200,     1200,      600,      120;
  6:    720,    4320,    10800,    14400,    10800,     4320,      720;
  7:   5040,   35280,   105840,   176400,   176400,   105840,    35280,     5040;
  8:  40320,  322560,  1128960,  2257920,  2822400,  2257920,  1128960,   322560,   40320;
  9: 362880, 3265920, 13063680, 30481920, 45722880, 45722880, 30481920, 13063680, 3265920, 362880;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+n*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[(n + n*x)*P[n - 1, 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] = (n + n*x)*v[i - 1]); 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-61ffea65f7eec286
  • 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-61ffea65f7eec286, e-83332bc01455e815, e-ab4aac9d5a3bd564, e-bf1e7898c429e3eb
  • 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-61ffea65f7eec286, e-83332bc01455e815, e-ab4aac9d5a3bd564, e-bf1e7898c429e3eb
  • 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-61ffea65f7eec286
  • 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-83332bc01455e815, e-bf1e7898c429e3eb
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-ab4aac9d5a3bd564

Proof chains

4 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 8 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