A113214 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,x\right)\,P_{n-2}(x) + \left(1 + 3\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	3														6
2	4	15	9													28
3	7	39	63	27												136
4	11	87	234	243	81											656
5	18	180	675	1161	891	243										3168
6	29	354	1710	4131	5184	3159	729									15296
7	47	675	3987	12447	21951	21627	10935	2187								73856
8	76	1257	8784	33669	76869	106191	86022	37179	6561							356608
9	123	2301	18567	84429	237168	424278	480411	330237	124659	19683						1721856
10	199	4155	38025	200151	668331	1472580	2157840	2066715	1233468	413343	59049					8313856
11	322	7422	75960	454356	1759239	4613355	8328825	10311705	8548983	4507407	1358127	177147				40142848
12	521	13140	148716	996462	4391091	13368645	28744470	43838415	46917711	34268103	16179426	4428675	531441			193826816
13	843	23091	286362	2124846	10502784	36432990	91019295	165370005	217917054	205175592	133864083	57218481	14348907	1594323		935878656
14	1364	40323	543771	4426542	24257799	94483260	269168670	568499715	892460025	1033947990	868374594	511777683	199821816	46235367	4782969	4518821888

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Technical details

Node id: n-d8e527df92e2a449 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fcdb948664301fc0

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 ?

A108051: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:   3,    3;
  2:   4,   15,     9;
  3:   7,   39,    63,    27;
  4:  11,   87,   234,   243,     81;
  5:  18,  180,   675,  1161,    891,    243;
  6:  29,  354,  1710,  4131,   5184,   3159,    729;
  7:  47,  675,  3987, 12447,  21951,  21627,  10935,   2187;
  8:  76, 1257,  8784, 33669,  76869, 106191,  86022,  37179,   6561;
  9: 123, 2301, 18567, 84429, 237168, 424278, 480411, 330237, 124659, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (1+3*x)*P(n-2,x) + (1+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 2, x] + (1 + 3*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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 2] + (1 + 3*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-849840c29c946dc7
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-849840c29c946dc7
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-849840c29c946dc7

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-849840c29c946dc7

Geometric basis map (elementary) polytool ?
- premise (sequence): Riordan array (1+2x,x(1+x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A111125 e-7b745496cf172799
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 1]
    
    no recorded property covers this requirement
  - theorem_check ma_wang_interval e-5a8f629e61d609b9 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
  - basis_map A038303 e-0902c3fe08444c8c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-e470a688ac88b1cf
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038255 e-2011332c4965ce14
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-defa6799aaae8d2a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038327 e-253047f151171d84
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-f8dba443f4d27d49
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038279 e-4738d2d3e61fb87a
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-fbfad365d270ad64
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027467 e-71209083fd4bb877
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-1b92f65567f189e7
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A007318 e-719615516772b132
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-fa99d43ba99a1c05
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038243 e-94eaee4d49f2668d
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-32ad399df63f844d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027466 e-9b73612a2d328481
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-921246387fcf3a1f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038231 e-a363e9242012bc21
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-85e6814e89c84b8b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038207 e-e5bdc121bd99907c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
    
    preserver_image affine, shift 0 e-e475554f8902db13
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A038291 e-e67dd89b1bcbf6c7
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-77c8319978ad9839
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (expanded above)
  - basis_map A027465 e-f9e5e2bc826d3dc0
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-f73e26dac50fbf3d
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1+2x,x(1+x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
    
    ↵ theorem_check ma_wang_interval e-5a8f629e61d609b9 (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