A257620: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(n) = 3*n + 3.

A257620 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A257620

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

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

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	3														6
2	9	36	9													54
3	27	297	297	27												648
4	81	2106	5346	2106	81											9720
5	243	13851	73386	73386	13851	243										174960
6	729	87480	868239	1761264	868239	87480	729									3674160
7	2187	540189	9388791	34158753	34158753	9388791	540189	2187								88179840
8	6561	3293622	95843088	578903274	1024762590	578903274	95843088	3293622	6561							2380855680
9	19683	19938879	941634720	8959544136	25791697782	25791697782	8959544136	941634720	19938879	19683						71425670400
10	59049	120223764	9013062213	130113762912	575025893586	928501120152	575025893586	130113762912	9013062213	120223764	59049					2357047123200
11	177147	723291201	84724272837	1804717834695	11748118713678	28788563928042	28788563928042	11748118713678	1804717834695	84724272837	723291201	177147				84853696435200
12	531441	4346124498	786387065166	24198342201450	224949162241935	800148999833028	1209119684977764	800148999833028	224949162241935	24198342201450	786387065166	4346124498	531441			3309294160972800
13	1594323	26097473187	7233944068422	316330879567878	4100187699672525	20476309377526749	44595089380525716	44595089380525716	20476309377526749	4100187699672525	316330879567878	7233944068422	26097473187	1594323		138990354760857600
14	4782969	156651800688	66123298070091	4056392541277728	71941734520827849	491579199785657232	1489357230184262259	2140564290265234368	1489357230184262259	491579199785657232	71941734520827849	4056392541277728	66123298070091	156651800688	4782969	6254565964238592000

Generated b-files: A034001 (row sums), A257620, A257620 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-dfedc6a36c97eba4 · revision 1 · JSON
Definition source: oeis_catalog:A257620
Fingerprint: f6-605cb6decd8257f5

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 ?

A034001: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
A257620: 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 = 0 ?
A257620: 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 = 0 ?
A052690: row sums, 13 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:     3,        3;
  2:     9,       36,         9;
  3:    27,      297,       297,         27;
  4:    81,     2106,      5346,       2106,          81;
  5:   243,    13851,     73386,      73386,       13851,         243;
  6:   729,    87480,    868239,    1761264,      868239,       87480,        729;
  7:  2187,   540189,   9388791,   34158753,    34158753,     9388791,     540189,      2187;
  8:  6561,  3293622,  95843088,  578903274,  1024762590,   578903274,   95843088,   3293622,     6561;
  9: 19683, 19938879, 941634720, 8959544136, 25791697782, 25791697782, 8959544136, 941634720, 19938879, 19683;

%F Formula

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

A257620: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     3,        3;
  2:     9,       36,         9;
  3:    27,      297,       297,         27;
  4:    81,     2106,      5346,       2106,          81;
  5:   243,    13851,     73386,      73386,       13851,         243;
  6:   729,    87480,    868239,    1761264,      868239,       87480,        729;
  7:  2187,   540189,   9388791,   34158753,    34158753,     9388791,     540189,      2187;
  8:  6561,  3293622,  95843088,  578903274,  1024762590,   578903274,   95843088,   3293622,     6561;
  9: 19683, 19938879, 941634720, 8959544136, 25791697782, 25791697782, 8959544136, 941634720, 19938879, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (3+3*n*x)*P(n-1,x) + (3*x-3*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[(3 + 3*n*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (3 + 3*n*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); 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.

A257620: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     3,        3;
  2:     9,       36,         9;
  3:    27,      297,       297,         27;
  4:    81,     2106,      5346,       2106,          81;
  5:   243,    13851,     73386,      73386,       13851,         243;
  6:   729,    87480,    868239,    1761264,      868239,       87480,        729;
  7:  2187,   540189,   9388791,   34158753,    34158753,     9388791,     540189,      2187;
  8:  6561,  3293622,  95843088,  578903274,  1024762590,   578903274,   95843088,   3293622,     6561;
  9: 19683, 19938879, 941634720, 8959544136, 25791697782, 25791697782, 8959544136, 941634720, 19938879, 19683;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+3*n*x)*P(n-1,x) + (3*x-3*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[(3 + 3*n*x)*P[n - 1, x] + (3*x - 3*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (3 + 3*n*x)*v[i - 1] + (3*x - 3*x^2)*deriv(v[i - 1], x)); 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-563d8c95cca8fabd
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-563d8c95cca8fabd, e-f71cdfb9d254d483
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-563d8c95cca8fabd, e-f71cdfb9d254d483
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-563d8c95cca8fabd
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-f71cdfb9d254d483

Proof chains

2 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

parent_provers root_domain e-f71cdfb9d254d483 root: theorem check

Parent-row provers of polytool polytool ?
theorem_check liu_wang e-563d8c95cca8fabd root: theorem check

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 3 proof steps

e-184795b7aaceffec gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) …
e-9a566c54793e3e13 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, …
e-9a566c54793e3e13 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, …

Related sequences ?

Same fingerprint (6)

Equal up to content, sign, reversal and a small index shift.

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