A099040 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 6\,x\right)\,P_{n-2}(x) + \left(2 + 6\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	6							8
2	6	30	36						72
3	16	120	288	216					640
4	44	432	1548	2376	1296				5696
5	120	1464	6984	16200	18144	7776			50688
6	328	4776	28440	88344	150336	132192	46656		451072
7	896	15168	108288	421632	964224	1290816	933120	279936	4014080

Show all 15 stored rows

Generated b-files: A057091 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-0892e4c474e4320d · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-3f5c634f6f648d33

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 ?

A057091: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 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:    2,      6;
  2:    6,     30,      36;
  3:   16,    120,     288,     216;
  4:   44,    432,    1548,    2376,     1296;
  5:  120,   1464,    6984,   16200,    18144,     7776;
  6:  328,   4776,   28440,   88344,   150336,   132192,    46656;
  7:  896,  15168,  108288,  421632,   964224,  1290816,   933120,   279936;
  8: 2448,  47232,  393120, 1840320,  5288976,  9533376, 10497600,  6438528,  1679616;
  9: 6688, 144864, 1377216, 7532352, 26078112, 59167584, 87806592, 82021248, 43670016, 10077696;

%F Formula

The row polynomials satisfy P(n,x) = (2+6*x)*P(n-2,x) + (2+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[(2 + 6*x)*P[n - 2, x] + (2 + 6*x)*P[n - 1, 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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 6*x)*v[i - 2] + (2 + 6*x)*v[i - 1]); 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-ab2091a27c0ed1a5
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-ab2091a27c0ed1a5
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-ab2091a27c0ed1a5

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

basis_map A013610 e-ab2091a27c0ed1a5

Geometric basis map (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 (-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 ?
  - 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 (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