A257611: Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1) and f(n) = 2*n + 3.

A257611 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 A257611

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

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

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	30	9													48
3	27	213	213	27												480
4	81	1308	2982	1308	81											5760
5	243	7431	32646	32646	7431	243										80640
6	729	40314	310263	587628	310263	40314	729									1290240
7	2187	212505	2695923	8701545	8701545	2695923	212505	2187								23224320
8	6561	1099704	22059036	113360904	191433990	113360904	22059036	1099704	6561							464486400
9	19683	5623179	173108220	1351133676	3579465642	3579465642	1351133676	173108220	5623179	19683						10218700800
10	59049	28529238	1318597941	15103042824	59641127202	93066106692	59641127202	15103042824	1318597941	28529238	59049					245248819200
11	177147	144004317	9829299585	160980746295	912804127230	2104476295026	2104476295026	912804127230	160980746295	9829299585	144004317	177147				6376469299200
12	531441	724450260	72117196386	1655242007940	13099479579135	42875861998248	63134288850780	42875861998248	13099479579135	1655242007940	72117196386	724450260	531441			178541140377600
13	1594323	3636600207	522931631202	16555873588338	178854357537225	806276317980789	1675903986731916	1675903986731916	806276317980789	178854357537225	16555873588338	522931631202	3636600207	1594323		5356234211328000
14	4782969	18229236402	3758709624003	162076153075092	2348183025441249	14237533642031982	40457809842613731	56980735548885144	40457809842613731	14237533642031982	2348183025441249	162076153075092	3758709624003	18229236402	4782969	171399494762496000

Generated b-files: A051578 (row sums), A257611, A257611 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-0c0e99d973615582 · revision 1 · JSON
Definition source: oeis_catalog:A257611
Fingerprint: f6-b446d021db3cc66d

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 ?

A051578: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
A052587: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A257611: 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 ?
A257611: 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 ?

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,      30,         9;
  3:    27,     213,       213,         27;
  4:    81,    1308,      2982,       1308,         81;
  5:   243,    7431,     32646,      32646,       7431,        243;
  6:   729,   40314,    310263,     587628,     310263,      40314,        729;
  7:  2187,  212505,   2695923,    8701545,    8701545,    2695923,     212505,      2187;
  8:  6561, 1099704,  22059036,  113360904,  191433990,  113360904,   22059036,   1099704,    6561;
  9: 19683, 5623179, 173108220, 1351133676, 3579465642, 3579465642, 1351133676, 173108220, 5623179, 19683;

%F Formula

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

A257611: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     3,       3;
  2:     9,      30,         9;
  3:    27,     213,       213,         27;
  4:    81,    1308,      2982,       1308,         81;
  5:   243,    7431,     32646,      32646,       7431,        243;
  6:   729,   40314,    310263,     587628,     310263,      40314,        729;
  7:  2187,  212505,   2695923,    8701545,    8701545,    2695923,     212505,      2187;
  8:  6561, 1099704,  22059036,  113360904,  191433990,  113360904,   22059036,   1099704,    6561;
  9: 19683, 5623179, 173108220, 1351133676, 3579465642, 3579465642, 1351133676, 173108220, 5623179, 19683;

%F Formula

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

A257611: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     3,       3;
  2:     9,      30,         9;
  3:    27,     213,       213,         27;
  4:    81,    1308,      2982,       1308,         81;
  5:   243,    7431,     32646,      32646,       7431,        243;
  6:   729,   40314,    310263,     587628,     310263,      40314,        729;
  7:  2187,  212505,   2695923,    8701545,    8701545,    2695923,     212505,      2187;
  8:  6561, 1099704,  22059036,  113360904,  191433990,  113360904,   22059036,   1099704,    6561;
  9: 19683, 5623179, 173108220, 1351133676, 3579465642, 3579465642, 1351133676, 173108220, 5623179, 19683;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+(2*n+1)*x)*P(n-1,x) + (2*x-2*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 + (2*n + 1)*x)*P[n - 1, x] + (2*x - 2*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 + (2*n + 1)*x)*v[i - 1] + (2*x - 2*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-4c1dd88e7fbb8285
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-4c1dd88e7fbb8285
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-4c1dd88e7fbb8285
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-4c1dd88e7fbb8285

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 e-4c1dd88e7fbb8285 root: theorem check

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

Used in 3 proof steps

e-2cce1f50097cb054 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,…
e-cf5b10dddbff9e73 obreschkoff_combination → obreschkoff_combination of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)…
e-cf5b10dddbff9e73 obreschkoff_combination → obreschkoff_combination of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)…

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