Real-rooted atlas

A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.

A090582 nonnegative coefficients

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

Recurrence from OEIS A090582
\(P_{n}(x) = \left(1 + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
266113
3243614175
4120240150301541
5720180015605406214683
65040151201680084001806126147293
7403201411201915201260004082457962541545835

Show all 15 stored rows

Generated b-files: A019538 (reversed triangle), A090582 ?

Computed in your browser ?

Technical details
Node id
n-3b2b9e31bd278a34 · revision 1 · JSON
Definition source
oeis_catalog:A090582
Fingerprint
f6-a83eeccfb5efd735

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:       2,        1;
  2:       6,        6,        1;
  3:      24,       36,       14,        1;
  4:     120,      240,      150,       30,        1;
  5:     720,     1800,     1560,      540,       62,       1;
  6:    5040,    15120,    16800,     8400,     1806,     126,      1;
  7:   40320,   141120,   191520,   126000,    40824,    5796,    254,     1;
  8:  362880,  1451520,  2328480,  1905120,   834120,  186480,  18150,   510,    1;
  9: 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+1+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[(n + 1 + 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] = (n + 1 + 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.

A019538: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,    2;
   3: 1,    6,     6;
   4: 1,   14,    36,     24;
   5: 1,   30,   150,    240,     120;
   6: 1,   62,   540,   1560,    1800,      720;
   7: 1,  126,  1806,   8400,   16800,    15120,     5040;
   8: 1,  254,  5796,  40824,  126000,   191520,   141120,    40320;
   9: 1,  510, 18150, 186480,  834120,  1905120,  2328480,  1451520,   362880;
  10: 1, 1022, 55980, 818520, 5103000, 16435440, 29635200, 30240000, 16329600, 3628800;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (n+(n-1)*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[(n + (n - 1)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], 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; for(n = 2, N, my(i = n); v[i] = (n + (n - 1)*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 = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A090582: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       14,        1;
   5:     120,      240,      150,       30,        1;
   6:     720,     1800,     1560,      540,       62,       1;
   7:    5040,    15120,    16800,     8400,     1806,     126,      1;
   8:   40320,   141120,   191520,   126000,    40824,    5796,    254,     1;
   9:  362880,  1451520,  2328480,  1905120,   834120,  186480,  18150,   510,    1;
  10: 3628800, 16329600, 30240000, 29635200, 16435440, 5103000, 818520, 55980, 1022, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+(n-1)*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[(n + (n - 1)*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] = (n + (n - 1)*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.

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-3e082ac2cc139530
  • 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-3e082ac2cc139530
  • 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-3e082ac2cc139530

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

Same fingerprint (5)

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