A028313: Elements in the 5-Pascal triangle (by row).

A028313 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A028313

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

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

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

\(P_{2}(x) = 1 + 5\,x + x^{2}\)

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	5	1													7
3	1	6	6	1												14
4	1	7	12	7	1											28
5	1	8	19	19	8	1										56
6	1	9	27	38	27	9	1									112
7	1	10	36	65	65	36	10	1								224
8	1	11	46	101	130	101	46	11	1							448
9	1	12	57	147	231	231	147	57	12	1						896
10	1	13	69	204	378	462	378	204	69	13	1					1792
11	1	14	82	273	582	840	840	582	273	82	14	1				3584
12	1	15	96	355	855	1422	1680	1422	855	355	96	15	1			7168
13	1	16	111	451	1210	2277	3102	3102	2277	1210	451	111	16	1		14336
14	1	17	127	562	1661	3487	5379	6204	5379	3487	1661	562	127	17	1	28672

b-files: A005009 (row sums), A028313, A028313 (reversed triangle) ?

Technical details

Node id: n-10c99b3e7ebd1948 · revision 0 · JSON
Definition source: oeis_catalog:A028313
Fingerprint: f6-61cf5fd362f41459

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

Matches ?

A028313 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?
A028313 reversed triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?

Row sums ? A005009, A135092, A125176

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  5,  1;
  3: 1,  6,  6,   1;
  4: 1,  7, 12,   7,   1;
  5: 1,  8, 19,  19,   8,   1;
  6: 1,  9, 27,  38,  27,   9,   1;
  7: 1, 10, 36,  65,  65,  36,  10,  1;
  8: 1, 11, 46, 101, 130, 101,  46, 11,  1;
  9: 1, 12, 57, 147, 231, 231, 147, 57, 12, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];
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; v[3] = 1 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1])/(n - 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)

polytool c48629374597

A028313 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  5,  1;
  3: 1,  6,  6,   1;
  4: 1,  7, 12,   7,   1;
  5: 1,  8, 19,  19,   8,   1;
  6: 1,  9, 27,  38,  27,   9,   1;
  7: 1, 10, 36,  65,  65,  36,  10,  1;
  8: 1, 11, 46, 101, 130, 101,  46, 11,  1;
  9: 1, 12, 57, 147, 231, 231, 147, 57, 12, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];
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; v[3] = 1 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1])/(n - 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)

polytool c48629374597

A028313 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  5,  1;
  3: 1,  6,  6,   1;
  4: 1,  7, 12,   7,   1;
  5: 1,  8, 19,  19,   8,   1;
  6: 1,  9, 27,  38,  27,   9,   1;
  7: 1, 10, 36,  65,  65,  36,  10,  1;
  8: 1, 11, 46, 101, 130, 101,  46, 11,  1;
  9: 1, 12, 57, 147, 231, 231, 147, 57, 12, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 5*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];
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; v[3] = 1 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1])/(n - 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers, Liu–Wang theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Liu–Wang theorem
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 3
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers

Proof chains ? JSON

Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Used in 5

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.