A099040 → D

nonnegative coefficients

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

Recurrence

\(\left(2\,x - n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-6 - 14\,x - 8\,x^{2} + 8\,n + 10\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 10\,x^{2} - 4\,x^{3} + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	2							2
2	2	8						10
3	0	16	24					40
4	0	8	72	64				144
5	0	0	72	256	160			488
6	0	0	24	384	800	384		1592
7	0	0	0	256	1600	2304	896	5056

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-5ec53b3631593330 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-6218d73036c1610d

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 ?

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: 2;
  2: 2,  8;
  3: 0, 16, 24;
  4: 0,  8, 72,  64;
  5: 0,  0, 72, 256,  160;
  6: 0,  0, 24, 384,  800,  384;
  7: 0,  0,  0, 256, 1600, 2304,   896;
  8: 0,  0,  0,  64, 1600, 5760,  6272,  2048;
  9: 0,  0,  0,   0,  800, 7680, 18816, 16384, 4608;

%F Formula

The row polynomials satisfy (n^2-n-(2*n-2)*x)*P(n,x) = -(2*n^2-8*n+6-(10*n-14)*x+8*x^2)*P(n-1,x) + ((4*n-4)*x+(2*n-10)*x^2-4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 8*n + 6 - (10*n - 14)*x + 8*x^2)*P[n - 1, x] + ((4*n - 4)*x + (2*n - 10)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n^2 - n - (2*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 8*n + 6 - (10*n - 14)*x + 8*x^2)*v[i - 1] + ((4*n - 4)*x + (2*n - 10)*x^2 - 4*x^3)*deriv(v[i - 1], x))/(n^2 - n - (2*n - 2)*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.

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-d51a9609918cc07c
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-d51a9609918cc07c
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-d51a9609918cc07c

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-d51a9609918cc07c

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Riordan array (1, 2+2x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A099040 e-52040eb0dce100d3
    
    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
  - theorem_check ma_wang_interval e-17786677d2ed021b root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
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    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-48e4dfa7d694f880
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038255 e-162624a94908f408
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-79e1a4a5e521405c
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A027466 e-3f03a2606d2e6662
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-5e00209341210844
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038303 e-532ccda39f04a0f0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-c2f106499058af8f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038207 e-72107609b2d5e6bc
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
    
    preserver_image affine, shift 0 e-8c4e4f825f4ab306
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038243 e-763de8a298cf812d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-bdc3e5aca0457f81
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038279 e-8232f658d776d13c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-4478346993b2c814
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A027465 e-af79ef336fd9f8da
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-319b2a76ea6c1ab0
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038291 e-c3d762863d6658d7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-b9371054cfe1d371
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A007318 e-e158703ce212fb71
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-dc7a132d665751c2
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038231 e-e36b44dfb8fba667
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-23d8b9b067cbe12f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)
  - basis_map A038327 e-ffeb0648278e3cee
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-1000187f132dd1ea
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 2+2x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
    
    ↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.