Real-rooted atlas

A100862: Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.

A100862 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A100862
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(-x + n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21315
3166114
41102110143
51155555151142
6121120215120211499
71282316656652312811850

Show all 15 stored rows

Generated b-files: A005425 (row sums), A100862, A100862 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(2\right)\,P_{n}(x) = \left(2 + 2\,x^{2} + n x - 2\,n x^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 6\,x^{2} + 2\,x^{3} - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-07e5356ad37f61ea · revision 2 · JSON
Definition source
oeis_catalog:A100862
derived:reversal (equal by e-efb682efd7da7c4c)
Fingerprint
f6-fae363dac50bb006

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): 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,  1;
  2: 1,  3,   1;
  3: 1,  6,   6,    1;
  4: 1, 10,  21,   10,    1;
  5: 1, 15,  55,   55,   15,    1;
  6: 1, 21, 120,  215,  120,   21,    1;
  7: 1, 28, 231,  665,  665,  231,   28,   1;
  8: 1, 36, 406, 1736, 2835, 1736,  406,  36,  1;
  9: 1, 45, 666, 3990, 9891, 9891, 3990, 666, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (n - 1)*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (n - 1)*x*v[i - 2]); 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.

A100862: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  6,   6,    1;
  4: 1, 10,  21,   10,    1;
  5: 1, 15,  55,   55,   15,    1;
  6: 1, 21, 120,  215,  120,   21,    1;
  7: 1, 28, 231,  665,  665,  231,   28,   1;
  8: 1, 36, 406, 1736, 2835, 1736,  406,  36,  1;
  9: 1, 45, 666, 3990, 9891, 9891, 3990, 666, 45, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (n - 1)*x*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (n - 1)*x*v[i - 2]); 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.

A100862: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   1;
  3: 1,  6,   6,    1;
  4: 1, 10,  21,   10,    1;
  5: 1, 15,  55,   55,   15,    1;
  6: 1, 21, 120,  215,  120,   21,    1;
  7: 1, 28, 231,  665,  665,  231,   28,   1;
  8: 1, 36, 406, 1736, 2835, 1736,  406,  36,  1;
  9: 1, 45, 666, 3990, 9891, 9891, 3990, 666, 45, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (n-1)*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (n - 1)*x*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + (n - 1)*x*v[i - 2]); 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-e7942008ad3e1420
  • 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-7078061f6d32a158, e-e7942008ad3e1420
  • 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-7078061f6d32a158, e-e7942008ad3e1420
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-e7942008ad3e1420
  • 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-7078061f6d32a158

Proof chains

3 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 13 proof steps

No candidate equalities or related sequences.

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