A128099 → D

nonnegative coefficients

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

Recurrence

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

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1								0
2	2							2
3	4							4
4	6	8						14
5	8	24						32
6	10	48	24					82
7	12	80	96					188
8	14	120	240	64				438
9	16	168	480	320				984
10	18	224	840	960	160			2202
11	20	288	1344	2240	960			4852
12	22	360	2016	4480	3360	384		10622
13	24	440	2880	8064	8960	2688		23056
14	26	528	3960	13440	20160	10752	896	49762

Generated b-files: A095977 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-f8f37b87137bc296 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-5410ddd5b5edb7cf

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 ?

A095977: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:  0;
  1:  0;
  2:  2;
  3:  4;
  4:  6,   8;
  5:  8,  24;
  6: 10,  48,  24;
  7: 12,  80,  96;
  8: 14, 120, 240,  64;
  9: 16, 168, 480, 320;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-1+(4*n-12)*x)*P(n-1,x) - (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 - 1 + (4*n - 12)*x)*P[n - 1, x] - (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 - 1 + (4*n - 12)*x)*v[i - 1] - (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-781b6acf5129326e
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-781b6acf5129326e
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-781b6acf5129326e

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-781b6acf5129326e

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - gamma_polynomial scale 1, shift 0 e-250eeed7ad07d148
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?
    
    premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
    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-1f8dc4b132ba6116 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
    
    preserver_image reversal, shift 0 e-8efc9aafb30c2198
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
    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-1f8dc4b132ba6116 (expanded above)
  - preserver_image reversal, shift 0 e-6b9d711da9b2e815
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    basis_map A027467 e-0db30c1d149f8e9a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-0ffa0091fcab81ab
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
    
    basis_map A038303 e-1078e1fe9c978d23
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-ba5819a43d999a58
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038327 e-39068ae2f1d878f6
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-0ef3f7736abad08b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    preserver_image reversal, shift 0 e-5066005127b21217
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pi…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ gamma_polynomial scale 1, shift 0 e-250eeed7ad07d148 (expanded above)
    
    parent_provers paper_theorem e-7c3667820056c57c root: theorem check
    
    Parent-row provers of polytool polytool ?
    
    basis_map A027465 e-76bf35985f9d5eb4
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-5f9e6216c74dfca3
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038243 e-7b55c4767302dd2a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-0c61b86c2e25257b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A007318 e-89ed187733702f47
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-38de1ebb8f2e73b6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038255 e-9c1dd42dfa910ea7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-4867241b67312c9e
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038207 e-a0ae771232cbba16
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
    
    preserver_image affine, shift 0 e-c345fe1d5254381d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038291 e-a2d6a4286b164d3d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-f1a9ca5f4baaa1dd
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038231 e-b4c854eb781c5167
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-b26782bfa4993a06
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A038279 e-c41010d0d9363851
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-765910d978fe3ce9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A027466 e-de853701eca80777
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-2f0380ac472e9edc
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): A Jacobsthal triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
    
    ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)
    
    basis_map A113953 e-4e53ceff1d4735da
    
    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-2d5b16bf2955a407 (expanded above)
  - ↵ parent_provers paper_theorem e-7c3667820056c57c (expanded above)

Related sequences ?

Same fingerprint (1)

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

A113953 → θ+1

References

[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
[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.