A099040: Riordan array (1, 2+2x).

A099040 nonnegative coefficients

Proved for all n: all roots real, in (−2, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A099040

\(P_{n}(x) = \left(2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	2	4						6
3	0	0	8	8					16
4	0	0	4	24	16				44
5	0	0	0	24	64	32			120
6	0	0	0	8	96	160	64		328
7	0	0	0	0	64	320	384	128	896

Show all 15 stored rows

Generated b-files: A099040 ?

Computed in your browser ?

Other recurrences

Basis-map coefficients of A099040, shifted · proved equal for all n (proof step)

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

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

Technical details

Node id: n-f5a001953ee28e8f · revision 1 · JSON
Definition source: oeis_catalog:A099040
derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A099040 (equal by e-09d9d4f57056315c)
Fingerprint: f6-b928c80a61c812bf

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: 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 ?

A002605: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A099040: triangle, 78 terms agree, 12 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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: 0, 2;
  2: 0, 2, 4;
  3: 0, 0, 8,  8;
  4: 0, 0, 4, 24, 16;
  5: 0, 0, 0, 24, 64,  32;
  6: 0, 0, 0,  8, 96, 160,   64;
  7: 0, 0, 0,  0, 64, 320,  384,  128;
  8: 0, 0, 0,  0, 16, 320,  960,  896,  256;
  9: 0, 0, 0,  0,  0, 160, 1280, 2688, 2048, 512;

%F Formula

The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + 2*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + 2*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.

A099040: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 2, 4;
  3: 0, 0, 8,  8;
  4: 0, 0, 4, 24, 16;
  5: 0, 0, 0, 24, 64,  32;
  6: 0, 0, 0,  8, 96, 160,   64;
  7: 0, 0, 0,  0, 64, 320,  384,  128;
  8: 0, 0, 0,  0, 16, 320,  960,  896,  256;
  9: 0, 0, 0,  0,  0, 160, 1280, 2688, 2048, 512;

%F Formula

The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] + 2*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 2*x*v[i - 1] + 2*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.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-2, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-52040eb0dce100d3
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-0a457ba83443136e, e-162624a94908f408, e-17786677d2ed021b, e-3f03a2606d2e6662, e-52040eb0dce100d3, e-532ccda39f04a0f0, e-72107609b2d5e6bc, e-763de8a298cf812d, e-8232f658d776d13c, e-af79ef336fd9f8da, e-c3d762863d6658d7, e-e158703ce212fb71, e-e36b44dfb8fba667, e-ffeb0648278e3cee
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-0a457ba83443136e, e-162624a94908f408, e-17786677d2ed021b, e-3f03a2606d2e6662, e-52040eb0dce100d3, e-532ccda39f04a0f0, e-72107609b2d5e6bc, e-763de8a298cf812d, e-8232f658d776d13c, e-af79ef336fd9f8da, e-c3d762863d6658d7, e-e158703ce212fb71, e-e36b44dfb8fba667, e-ffeb0648278e3cee
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-17786677d2ed021b
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-17786677d2ed021b
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-0a457ba83443136e, e-162624a94908f408, e-3f03a2606d2e6662, e-532ccda39f04a0f0, e-72107609b2d5e6bc, e-763de8a298cf812d, e-8232f658d776d13c, e-af79ef336fd9f8da, e-c3d762863d6658d7, e-e158703ce212fb71, e-e36b44dfb8fba667, e-ffeb0648278e3cee

Proof chains

15 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

basis_map A027467 e-0a457ba83443136e

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 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
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 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
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)
definition_equality e-09d9d4f57056315c root: theorem check

Ore-algebra identity of two definitions polytool ?
↵ theorem_check ma_wang_interval e-17786677d2ed021b (expanded above)

Used in 26 proof steps

e-23c3ede392db8800 basis_map (A007318) → basis_map_A007318 of Riordan array (1, 2+2x).
e-76d46a3bbdce7b63 basis_map (A013609) → basis_map_A013609 of Riordan array (1, 2+2x).
e-a2850f8f2b718783 basis_map (A008277) → basis_map_A008277 of Riordan array (1, 2+2x).
e-ab2091a27c0ed1a5 basis_map (A013610) → basis_map_A013610 of Riordan array (1, 2+2x).
e-da161a9cde75b31f basis_map (A013611) → basis_map_A013611 of Riordan array (1, 2+2x).
e-1000187f132dd1ea preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-23d8b9b067cbe12f preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-319b2a76ea6c1ab0 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-4478346993b2c814 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-48e4dfa7d694f880 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-5e00209341210844 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-79e1a4a5e521405c preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-86e33cb6427ca24b preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Riordan array (1, 2+2x).
e-8bdfc53769ec9007 preserver_image (affine, shift 0) → basis_map_A007318 of Riordan array (1, 2+2x).
e-8c4e4f825f4ab306 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-974408a4688a73a4 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Riordan array (1, 2+2x).
e-9bce1221c4367de4 preserver_image (reversal, shift 0) → gamma_polynomial of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*…
e-a0f85d448207b69e preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Riordan array (1, 2+2x).
e-b9371054cfe1d371 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-bdc3e5aca0457f81 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-c2f106499058af8f preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-d427c42eda759629 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Riordan array (1, 2+2x).
e-d51a9609918cc07c preserver_image (derivative_polynomial, shift 0) → derivative of Riordan array (1, 2+2x).
e-dc7a132d665751c2 preserver_image (affine, shift 0) → basis-map coefficients of Riordan array (1, 2+2x).
e-07e54b52255e06bd veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Riordan array (1, 2+2x).
e-3006d34f722f9e52 veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Riordan array (1, 2+2x).

Related sequences ?

Same fingerprint (1)

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

A059473 → γ

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

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	2	4						6
3	0	0	8	8					16
4	0	0	4	24	16				44
5	0	0	0	24	64	32			120
6	0	0	0	8	96	160	64		328
7	0	0	0	0	64	320	384	128	896

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	2	4						6
3	0	0	8	8					16
4	0	0	4	24	16				44
5	0	0	0	24	64	32			120
6	0	0	0	8	96	160	64		328
7	0	0	0	0	64	320	384	128	896

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	2	4						6
3	0	0	8	8					16
4	0	0	4	24	16				44
5	0	0	0	24	64	32			120
6	0	0	0	8	96	160	64		328
7	0	0	0	0	64	320	384	128	896