A113214 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(5 + x - 2\,n\right)\,P_{n-1}(x) + \left(5 + 6\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

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	1														4
2	4	5	1													10
3	7	13	7	1												28
4	11	29	26	9	1											76
5	18	60	75	43	11	1										208
6	29	118	190	153	64	13	1									568
7	47	225	443	461	271	89	15	1								1552
8	76	419	976	1247	949	437	118	17	1							4240
9	123	767	2063	3127	2928	1746	659	151	19	1						11584
10	199	1385	4225	7413	8251	6060	2960	945	188	21	1					31648
11	322	2474	8440	16828	21719	18985	11425	4715	1303	229	23	1				86464
12	521	4380	16524	36906	54211	55015	39430	20045	7151	1741	274	25	1			236224
13	843	7697	31818	78698	129664	149930	124855	75615	33214	10424	2267	323	27	1		645376
14	1364	13441	60419	163946	299479	388820	369230	259945	136025	52530	14706	2889	376	29	1	1763200

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Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

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

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

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

Technical details

Node id: n-fb2cd7d8c683bee9 · revision 1 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-0b47fc86f50f7cc5)
Fingerprint: f6-a33aa4f865b92cc2

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 ?

A026150: 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,   1;
  2:   4,   5,    1;
  3:   7,  13,    7,    1;
  4:  11,  29,   26,    9,    1;
  5:  18,  60,   75,   43,   11,    1;
  6:  29, 118,  190,  153,   64,   13,   1;
  7:  47, 225,  443,  461,  271,   89,  15,   1;
  8:  76, 419,  976, 1247,  949,  437, 118,  17,  1;
  9: 123, 767, 2063, 3127, 2928, 1746, 659, 151, 19, 1;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n-5-x)*P(n-1,x) + (5+6*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 5 - x)*P[n - 1, x] + (5 + 6*x + x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 5 - x)*v[i - 1] + (5 + 6*x + x^2)*deriv(v[i - 1], x))/(n)); 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, -1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-651850b9e78a2c19
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-48de33f15b8f0da9, e-651850b9e78a2c19
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-48de33f15b8f0da9, e-651850b9e78a2c19
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-48de33f15b8f0da9

Proof chains

3 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 A007318 e-48de33f15b8f0da9

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)
definition_equality e-0b47fc86f50f7cc5 root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image affine, shift 0 e-651850b9e78a2c19

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 (-inf, 0]

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