Real-rooted atlas

A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).

A019538 nonnegative coefficients

Proved for all n: all roots real, in (−1, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A019538
\(P_{n}(x) = \left(1 + 2\,x\right)\,P_{n-1}(x) + \left(x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
316613
4114362475
5130150240120541
6162540156018007204683
71126180684001680015120504047293
8125457964082412600019152014112040320545835

Show all 15 stored rows

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

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-d938c1937a93c4df · revision 2 · JSON
Definition source
oeis_catalog:A019538
derived:mobius_x_over_x_plus_one (equal by e-4bbc13aea32d7e48)
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 -1)
  • 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 = 1, …, ascending powers of x
%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
The row polynomials satisfy P(n,x) = (1+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*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {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*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A019538: triangle, offset 1, from the row P1
%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
The row polynomials satisfy P(n,x) = (1+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*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*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.

A090582: reversed triangle, offset 1, from the row P1
%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
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+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*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] = (1 + 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 = Vecrev(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 >= 1, is zero or real-rooted with all roots in (-1, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-e395982737c903b7
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-052cfd8551fd4fcd, e-b7f464e4dcf953bf, e-e395982737c903b7
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-052cfd8551fd4fcd, e-b7f464e4dcf953bf, e-e395982737c903b7
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-e395982737c903b7
  • Real-rooted. Every row P_n, n >= 1, 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-052cfd8551fd4fcd
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-1, 0]: source [-1, 1], target [-1, 0]
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-52b4a42a5c10d66a
  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-b7f464e4dcf953bf

Proof chains

5 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 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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741