A011973 → ⊙ C(n,k) → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-12\,x + n + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 0\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
0				0
1				0
2	2			2
3	6			6
4	12	12		24
5	20	60		80
6	30	180	60	270
7	42	420	420	882

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-61c7ae27dda08fe6 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-f68b3fe345bb2270

Observed on the stored rows ?

Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
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 ?

No OEIS matches.

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:  0;
  1:  0;
  2:  2;
  3:  6;
  4: 12,   12;
  5: 20,   60;
  6: 30,  180,   60;
  7: 42,  420,  420;
  8: 56,  840, 1680,  280;
  9: 72, 1512, 5040, 2520;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n+(4*n-12)*x)*P(n-1,x) + (2*x-8*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + (4*n - 12)*x)*P[n - 1, x] + (2*x - 8*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((n + (4*n - 12)*x)*v[i - 1] + (2*x - 8*x^2)*deriv(v[i - 1], x))/(n - 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.

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-702433e45b2ced1d
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-702433e45b2ced1d
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-702433e45b2ced1d

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

preserver_image derivative_polynomial, shift 0 e-702433e45b2ced1d

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image euler_polynomial, shift 0 e-14cbfe8df545ebf2
    
    Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): Triangular array of Motzkin polynomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    gamma_polynomial scale 1, shift 1 e-3261655b8c5a1522
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): narayana
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
    
    preserver_image reversal, shift 0 e-29ef477df4172e22
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): narayana
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)
    
    gamma_polynomial scale 1, shift 0 e-55e5f4941dd58e15
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-d285019b29e04549 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
    
    preserver_image reversal, shift 0 e-14da04f2a2dce119
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-d285019b29e04549 (expanded above)
    
    preserver_image hadamard_multiplier, shift 0 e-e05d1c749616842f
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): delannoy
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
    
    preserver_image reversal, shift 0 e-266d85ca03a298fa
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): delannoy
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
    
    basis_map A008288 e-e771e35dc9ee3fbd
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    basis_map A008288 e-33dba7e79fe9caa8
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
    
    ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
    
    ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)
  - preserver_image hadamard_multiplier, shift 0 e-2126fae20f2e5116
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ parent_provers paper_theorem e-0801036cdfff24d2 (expanded above)
    
    ↵ parent_provers paper_theorem e-0d3631d0f359c9ce (expanded above)
    
    ↵ gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25 (expanded above)
  - gamma_polynomial scale 1, shift 0 e-e4032a9809c053c3
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Square the entries of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-c6b04c4560617585 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
    
    preserver_image reversal, shift 0 e-1906182487ade7a7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Square the entries of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check three_term_recurrence e-c6b04c4560617585 (expanded above)

Related sequences ?

Same fingerprint (4)

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

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023