A144443: Triangle read by rows: T(n, k) = (6n-6k+1)T(n-1, k-1) + (6k-5)T(n-1, k) + 6T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A144443 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A144443

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

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

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

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	20	1													22
3	1	159	159	1												320
4	1	1138	4254	1138	1											6532
5	1	7997	77878	77878	7997	1										171752
6	1	56016	1219167	2984888	1219167	56016	1									5535256
7	1	392155	17633649	87659315	87659315	17633649	392155	1								211370240
8	1	2745134	244083268	2219485106	4400875078	2219485106	244083268	2745134	1							9333502096
9	1	19215993	3293476176	51307099824	179351871126	179351871126	51307099824	3293476176	19215993	1						467943326240
10	1	134512012	43773244749	1117918871832	6395476382274	11146221260280	6395476382274	1117918871832	43773244749	134512012	1					26260827282016
11	1	941584151	576565638355	23405108414565	208265263644570	583241596439574	583241596439574	208265263644570	23405108414565	576565638355	941584151	1				1630978951442432
12	1	6591089130	7553597003898	476670809454754	6360189416658927	27074268684636948	43226755464090156	27074268684636948	6360189416658927	476670809454754	7553597003898	6591089130	1			111064133661777472
13	1	46137623989	98644013527290	9520974190708234	185362060586972035	1152201202221900231	2767082955189361980	2767082955189361980	1152201202221900231	185362060586972035	9520974190708234	98644013527290	46137623989	1		8228531764680187520
14	1	322963368008	1285779768940747	187552980111808264	5217690965164231673	45951311737662322648	159002373862987326267	238228494679069671216	159002373862987326267	45951311737662322648	5217690965164231673	187552980111808264	1285779768940747	322963368008	1	658948925976385666432

Generated b-files: A144443, A144443 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-a0dd3d6ef57da7a5 · revision 1 · JSON
Definition source: oeis_catalog:A144443
Fingerprint: f6-d813e141c637fb10

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 ?

A144443: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
A144443: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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,       20,          1;
  3: 1,      159,        159,           1;
  4: 1,     1138,       4254,        1138,            1;
  5: 1,     7997,      77878,       77878,         7997,            1;
  6: 1,    56016,    1219167,     2984888,      1219167,        56016,           1;
  7: 1,   392155,   17633649,    87659315,     87659315,     17633649,      392155,          1;
  8: 1,  2745134,  244083268,  2219485106,   4400875078,   2219485106,   244083268,    2745134,        1;
  9: 1, 19215993, 3293476176, 51307099824, 179351871126, 179351871126, 51307099824, 3293476176, 19215993, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(6*n-5)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) + 6*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 + (6*n - 5)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], x] + 6*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 + (6*n - 5)*x)*v[i - 1] + (6*x - 6*x^2)*deriv(v[i - 1], x) + 6*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.

A144443: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       20,          1;
   4: 1,      159,        159,           1;
   5: 1,     1138,       4254,        1138,            1;
   6: 1,     7997,      77878,       77878,         7997,            1;
   7: 1,    56016,    1219167,     2984888,      1219167,        56016,           1;
   8: 1,   392155,   17633649,    87659315,     87659315,     17633649,      392155,          1;
   9: 1,  2745134,  244083268,  2219485106,   4400875078,   2219485106,   244083268,    2745134,        1;
  10: 1, 19215993, 3293476176, 51307099824, 179351871126, 179351871126, 51307099824, 3293476176, 19215993, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(6*n-11)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) + 6*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n - 11)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], x] + 6*x*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (6*n - 11)*x)*v[i - 1] + (6*x - 6*x^2)*deriv(v[i - 1], x) + 6*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A144443: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,        1;
   3: 1,       20,          1;
   4: 1,      159,        159,           1;
   5: 1,     1138,       4254,        1138,            1;
   6: 1,     7997,      77878,       77878,         7997,            1;
   7: 1,    56016,    1219167,     2984888,      1219167,        56016,           1;
   8: 1,   392155,   17633649,    87659315,     87659315,     17633649,      392155,          1;
   9: 1,  2745134,  244083268,  2219485106,   4400875078,   2219485106,   244083268,    2745134,        1;
  10: 1, 19215993, 3293476176, 51307099824, 179351871126, 179351871126, 51307099824, 3293476176, 19215993, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(6*n-11)*x)*P(n-1,x) + (6*x-6*x^2)*d/dx P(n-1,x) + 6*x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n - 11)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], x] + 6*x*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + (6*n - 11)*x)*v[i - 1] + (6*x - 6*x^2)*deriv(v[i - 1], x) + 6*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-bfe3673616f49c96
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-bfe3673616f49c96
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-bfe3673616f49c96
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-bfe3673616f49c96

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

theorem_check liu_wang_derivative_lag e-bfe3673616f49c96 root: theorem check

Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Used in 1 proof step

e-b469c0e3f65675eb gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(…

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