A113953 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 6\,x\right)\,P_{n-2}(x) + \left(1 + 3\,x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	3														4
2	3	12	9													24
3	5	33	63	27												128
4	11	90	252	270	81											704
5	21	219	846	1458	1053	243										3840
6	43	528	2547	6048	7209	3888	729									20992
7	85	1221	7137	21681	36207	32319	13851	2187								114688
8	171	2790	19062	70470	151956	191970	135594	48114	6561							626688
9	341	6255	49032	213840	565866	929718	933120	542376	164025	19683						3424256
10	683	13884	122661	616248	1934118	3922992	5145282	4251528	2092959	551124	59049					18710528
11	1365	30489	299907	1706103	6197634	14979978	24358806	26370846	18429849	7853517	1830519	177147				102236160
12	2731	66450	720000	4574286	18881667	53023572	103127256	138822012	127237473	76803066	28815912	6022998	531441			558628864
13	5461	143811	1702098	11945934	55236411	176814333	400795452	647098308	738788283	584801613	310007250	103808142	19663317	1594323		3052404736
14	10923	309480	3972231	30520800	156283263	561860712	1455634395	2745892224	3767490225	3718197432	2582862309	1218771360	368288613	63772920	4782969	16678649856

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Technical details

Node id: n-ad5020a0c2e8a31d · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-354f21f96faadf4a

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 ?

A180222: 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:   1,    3;
  2:   3,   12,     9;
  3:   5,   33,    63,     27;
  4:  11,   90,   252,    270,     81;
  5:  21,  219,   846,   1458,   1053,    243;
  6:  43,  528,  2547,   6048,   7209,   3888,    729;
  7:  85, 1221,  7137,  21681,  36207,  32319,  13851,   2187;
  8: 171, 2790, 19062,  70470, 151956, 191970, 135594,  48114,   6561;
  9: 341, 6255, 49032, 213840, 565866, 929718, 933120, 542376, 164025, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (2+6*x)*P(n-2,x) + (1+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 6*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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 6*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-900137e7780f9cba
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-900137e7780f9cba
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-900137e7780f9cba

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-900137e7780f9cba

Geometric basis map (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
    
    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 (expanded above)
    
    ↵ basis_map A038303 e-1078e1fe9c978d23 (expanded above)
    
    ↵ basis_map A038327 e-39068ae2f1d878f6 (expanded above)
    
    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 root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ basis_map A027465 e-76bf35985f9d5eb4 (expanded above)
  - ↵ basis_map A038243 e-7b55c4767302dd2a (expanded above)
  - ↵ basis_map A007318 e-89ed187733702f47 (expanded above)
  - ↵ basis_map A038255 e-9c1dd42dfa910ea7 (expanded above)
  - ↵ basis_map A038207 e-a0ae771232cbba16 (expanded above)
  - ↵ basis_map A038291 e-a2d6a4286b164d3d (expanded above)
  - ↵ basis_map A038231 e-b4c854eb781c5167 (expanded above)
  - ↵ basis_map A038279 e-c41010d0d9363851 (expanded above)
  - ↵ basis_map A027466 e-de853701eca80777 (expanded above)
  - ↵ basis_map A113953 e-4e53ceff1d4735da (expanded above)
  - ↵ theorem_check ma_wang_interval e-2d5b16bf2955a407 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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