A144441: Triangle T(n,k) read by rows: T(n, k) = (4n-4k+1)T(n-1, k-1) + (4k-3)T(n-1, k) + 4T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

A144441 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 A144441

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

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

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

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	14	1													16
3	1	83	83	1												168
4	1	432	1550	432	1											2416
5	1	2181	19898	19898	2181	1										44160
6	1	10930	217887	523548	217887	10930	1									981184
7	1	54679	2199237	10589795	10589795	2199237	54679	1								25687424
8	1	273428	21203828	184722860	362147222	184722860	21203828	273428	1							774547456
9	1	1367177	198982580	2940289828	10078042014	10078042014	2940289828	198982580	1367177	1						26437363200
10	1	6835926	1837053773	44079077896	245572851378	424726353476	245572851378	44079077896	1837053773	6835926	1					1007717991424
11	1	34179675	16791881927	634446717477	5464792891722	15098886875502	15098886875502	5464792891722	634446717477	16791881927	34179675	1				42429905092608
12	1	170898424	152555647722	8876455173592	114014537147599	476537909650992	756643249189004	476537909650992	114014537147599	8876455173592	152555647722	170898424	1			1955806506225664
13	1	854492173	1380827977278	121715866341006	2269213759801995	13791635000108487	32796076157105876	32796076157105876	13791635000108487	2269213759801995	121715866341006	1380827977278	854492173	1		97960044931653632
14	1	4272460922	12470005505675	1645053744001476	43602490257309529	374041302263542438	1276932010569830939	1905198990108896824	1276932010569830939	374041302263542438	43602490257309529	1645053744001476	12470005505675	4272460922	1	5297665652334198784

Generated b-files: A144441, A144441 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-723ec2078be12613 · revision 1 · JSON
Definition source: oeis_catalog:A144441
Fingerprint: f6-7729bd9220ade5d3

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 ?

A144441: 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 ?
A144441: 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,      14,         1;
  3: 1,      83,        83,          1;
  4: 1,     432,      1550,        432,           1;
  5: 1,    2181,     19898,      19898,        2181,           1;
  6: 1,   10930,    217887,     523548,      217887,       10930,          1;
  7: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;
  8: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;
  9: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-3)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*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 + (4*n - 3)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*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 + (4*n - 3)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*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.

A144441: triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,       1;
   3: 1,      14,         1;
   4: 1,      83,        83,          1;
   5: 1,     432,      1550,        432,           1;
   6: 1,    2181,     19898,      19898,        2181,           1;
   7: 1,   10930,    217887,     523548,      217887,       10930,          1;
   8: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;
   9: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;
  10: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*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 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*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 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*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.

A144441: reversed triangle, offset 1, from the row P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,       1;
   3: 1,      14,         1;
   4: 1,      83,        83,          1;
   5: 1,     432,      1550,        432,           1;
   6: 1,    2181,     19898,      19898,        2181,           1;
   7: 1,   10930,    217887,     523548,      217887,       10930,          1;
   8: 1,   54679,   2199237,   10589795,    10589795,     2199237,      54679,         1;
   9: 1,  273428,  21203828,  184722860,   362147222,   184722860,   21203828,    273428,       1;
  10: 1, 1367177, 198982580, 2940289828, 10078042014, 10078042014, 2940289828, 198982580, 1367177, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(4*n-7)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) + 4*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 + (4*n - 7)*x)*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x] + 4*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 + (4*n - 7)*x)*v[i - 1] + (4*x - 4*x^2)*deriv(v[i - 1], x) + 4*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-494af060e65fc342
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-494af060e65fc342
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-494af060e65fc342
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-494af060e65fc342

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-494af060e65fc342 root: theorem check

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

Used in 1 proof step

e-24cd925870a5c300 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*…

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