A081580: Pascal-(1,5,1) array.

A081580 palindromic γ-positive nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A081580

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

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

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

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	1														2
2	1	7	1													9
3	1	13	13	1												28
4	1	19	61	19	1											101
5	1	25	145	145	25	1										342
6	1	31	265	595	265	31	1									1189
7	1	37	421	1585	1585	421	37	1								4088
8	1	43	613	3331	6145	3331	613	43	1							14121
9	1	49	841	6049	17401	17401	6049	841	49	1						48682
10	1	55	1105	9955	40105	65527	40105	9955	1105	55	1					167969
11	1	61	1405	15265	80305	192637	192637	80305	15265	1405	61	1				579348
12	1	67	1741	22195	145345	473467	712909	473467	145345	22195	1741	67	1			1998541
13	1	73	2113	30961	243865	1020337	2149561	2149561	1020337	243865	30961	2113	73	1		6893822
14	1	79	2521	41779	385801	1990927	5537233	7863667	5537233	1990927	385801	41779	2521	79	1	23780349

Generated b-files: A081580, A081580 (reversed triangle) ?

Computed in your browser ?

Other recurrences

Reversed rows · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(x^{2} + n - 5\,n x - n x^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(33\,x^{2} + 3\,x^{3} - n x - 22\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x) + \left(x^{2} + 22\,x^{3} + x^{4}\right)\,P_{n-1}''(x)\)

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

Technical details

Node id: n-6ad01dc6e705cdb6 · revision 2 · JSON
Definition source: oeis_catalog:A081580
derived:reversal (equal by e-cd4ca169930de37f)
Fingerprint: f6-267d5e5110f3ca83

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A081580: triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A081580: reversed triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A002532: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 1,  1;
  2: 1,  7,   1;
  3: 1, 13,  13,    1;
  4: 1, 19,  61,   19,     1;
  5: 1, 25, 145,  145,    25,     1;
  6: 1, 31, 265,  595,   265,    31,    1;
  7: 1, 37, 421, 1585,  1585,   421,   37,   1;
  8: 1, 43, 613, 3331,  6145,  3331,  613,  43,  1;
  9: 1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 5*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 5*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.

A081580: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  7,   1;
  3: 1, 13,  13,    1;
  4: 1, 19,  61,   19,     1;
  5: 1, 25, 145,  145,    25,     1;
  6: 1, 31, 265,  595,   265,    31,    1;
  7: 1, 37, 421, 1585,  1585,   421,   37,   1;
  8: 1, 43, 613, 3331,  6145,  3331,  613,  43,  1;
  9: 1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 5*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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 5*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.

A081580: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  7,   1;
  3: 1, 13,  13,    1;
  4: 1, 19,  61,   19,     1;
  5: 1, 25, 145,  145,    25,     1;
  6: 1, 31, 265,  595,   265,    31,    1;
  7: 1, 37, 421, 1585,  1585,   421,   37,   1;
  8: 1, 43, 613, 3331,  6145,  3331,  613,  43,  1;
  9: 1, 49, 841, 6049, 17401, 17401, 6049, 841, 49, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 5*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 5*x*P[n - 2, x]];
Table[Reverse[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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + 5*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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 (-inf, 0)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-307703197d4a988b
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-307703197d4a988b, e-bf7624d7e60440c3
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-307703197d4a988b, e-bf7624d7e60440c3
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-307703197d4a988b
Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-5 - sqrt(30), -5 + sqrt(30)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-5 - sqrt(30), -5 + sqrt(30)] has a real-rooted image
The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.

by the cited steps: e-e353876edd912a5b
Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-5 - sqrt(30), -5 + sqrt(30)] to roots in [-11 - 2 sqrt(30), -11 + 2 sqrt(30)]: every real polynomial with all roots in E = [-5 - sqrt(30), -5 + sqrt(30)] has an image with all roots in I = [-11 - 2 sqrt(30), -11 + 2 sqrt(30)]
The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.

by the cited steps: e-e353876edd912a5b
Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-5 - sqrt(30), 0] to roots in [-11 - 2 sqrt(30), -11 + 2 sqrt(30)]: every PF input with all roots in [-5 - sqrt(30), 0] has a PF image; no larger [-R, 0] works
The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.

by the cited steps: e-e353876edd912a5b
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-bf7624d7e60440c3

Proof chains

4 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_theorem e-e353876edd912a5b root: theorem check

Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
definition_equality e-cd4ca169930de37f root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image reversal, shift 0 e-bf7624d7e60440c3

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Pascal-(1,5,1) array.
  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-307703197d4a988b root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
↵ theorem_check liu_wang_derivative_lag e-307703197d4a988b (expanded above)

Used in 14 proof steps

e-1ae1e4b2be227437 basis_map (A007318) → basis_map_A007318 of Pascal-(1,5,1) array.
e-34022a8b1ec61d7e basis_map (A008277) → basis_map_A008277 of Pascal-(1,5,1) array.
e-4648c18632cda38e basis_map (A013611) → basis_map_A013611 of Pascal-(1,5,1) array.
e-8d949030857bf4e6 basis_map (A013610) → basis_map_A013610 of Pascal-(1,5,1) array.
e-aa38063cc8d845b3 basis_map (A013609) → basis_map_A013609 of Pascal-(1,5,1) array.
e-90e54edd477f164c gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Pascal-(1,5,1) array.
e-28cdd807988aaaf4 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Pascal-(1,5,1) array.
e-5726a07dd01f8b02 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Pascal-(1,5,1) array.
e-5eeadacb5f413009 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Pascal-(1,5,1) array.
e-b939e0864e65ea75 preserver_image (derivative_polynomial, shift 0) → derivative of Pascal-(1,5,1) array.
e-bf7624d7e60440c3 preserver_image (reversal, shift 0) → Pascal-(1,5,1) array.
e-dd36dc372b4d6491 preserver_image (affine, shift 0) → basis_map_A007318 of Pascal-(1,5,1) array.
e-7595c55a8f8b4fd6 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Pascal-(1,5,1) array.
e-7dc5699a1f5c04c0 veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Pascal-(1,5,1) array.

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