Real-rooted atlas

A008517: Second-order Eulerian triangle T(n,k), 1 <= k <= n.

A008517 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 A008517
\(P_{n}(x) = \left(1 + 2\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
218615
31225824105
4152328444120945
5111414524400370872010395
6124056103212058140339845040135135
7149419950195800644020785304341136403202027025
81100467260106250057655001244006411026296373392036288034459425
912026218848532616044765000155357384238904904162186912443390403628800654729075
10140726950382524390431436972016483843044002695088464216395225074812165683564803991680013749310575
111816621706261148763762051482776155489607845604139878410118043302492199790224407883018247827719040479001600316234143225
1211635666996965072592761266981777613432342022468772004638418181886423042549865473424188307966182469792943680011533608576062270208007905853580625
1313273820507988218982980875016052228108467651241676348321493922829991006611257494373464592647283751398723968957805580812550904017920181099255680087178291200213458046676875
141655046240789092925269204298260063408308444327968783913848313123943655878155201111747472569680179717122069056016664244862714568650232532195842369082715430403019637698560013076743680006190283353629375

Generated b-files: A008517, A112007 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-d4c21302c908a2cf · revision 1 · JSON
Definition source
oeis_catalog:A008517
Fingerprint
f6-36df5d4b0f5901bf

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: no
  • γ-positive (gamma_positive): no
  • 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,    2;
  2: 1,    8,      6;
  3: 1,   22,     58,      24;
  4: 1,   52,    328,     444,      120;
  5: 1,  114,   1452,    4400,     3708,       720;
  6: 1,  240,   5610,   32120,    58140,     33984,      5040;
  7: 1,  494,  19950,  195800,   644020,    785304,    341136,     40320;
  8: 1, 1004,  67260, 1062500,  5765500,  12440064,  11026296,   3733920,   362880;
  9: 1, 2026, 218848, 5326160, 44765000, 155357384, 238904904, 162186912, 44339040, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*n*x)*P(n-1,x) + (x-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[(1 + 2*n*x)*P[n - 1, x] + (x - 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] = (1 + 2*n*x)*v[i - 1] + (x - 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.

A008517: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,    2;
   3: 1,    8,      6;
   4: 1,   22,     58,      24;
   5: 1,   52,    328,     444,      120;
   6: 1,  114,   1452,    4400,     3708,       720;
   7: 1,  240,   5610,   32120,    58140,     33984,      5040;
   8: 1,  494,  19950,  195800,   644020,    785304,    341136,     40320;
   9: 1, 1004,  67260, 1062500,  5765500,  12440064,  11026296,   3733920,   362880;
  10: 1, 2026, 218848, 5326160, 44765000, 155357384, 238904904, 162186912, 44339040, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], 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; for(n = 2, N, my(i = n); v[i] = (1 + (2*n - 2)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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.

A112007: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       2,        1;
  2:       6,        8,         1;
  3:      24,       58,        22,         1;
  4:     120,      444,       328,        52,         1;
  5:     720,     3708,      4400,      1452,       114,        1;
  6:    5040,    33984,     58140,     32120,      5610,      240,       1;
  7:   40320,   341136,    785304,    644020,    195800,    19950,     494,      1;
  8:  362880,  3733920,  11026296,  12440064,   5765500,  1062500,   67260,   1004,    1;
  9: 3628800, 44339040, 162186912, 238904904, 155357384, 44765000, 5326160, 218848, 2026, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*n*x)*P(n-1,x) + (x-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[(1 + 2*n*x)*P[n - 1, x] + (x - 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] = (1 + 2*n*x)*v[i - 1] + (x - 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-83880dab33818290
  • 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-83880dab33818290
  • 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-83880dab33818290
  • 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-83880dab33818290

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 2 proof steps

Same fingerprint (5)

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