A134059: Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.

A134059 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A134059

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

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

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	3														6
2	3	6	3													12
3	3	9	9	3												24
4	3	12	18	12	3											48
5	3	15	30	30	15	3										96
6	3	18	45	60	45	18	3									192
7	3	21	63	105	105	63	21	3								384
8	3	24	84	168	210	168	84	24	3							768
9	3	27	108	252	378	378	252	108	27	3						1536
10	3	30	135	360	630	756	630	360	135	30	3					3072
11	3	33	165	495	990	1386	1386	990	495	165	33	3				6144
12	3	36	198	660	1485	2376	2772	2376	1485	660	198	36	3			12288
13	3	39	234	858	2145	3861	5148	5148	3861	2145	858	234	39	3		24576
14	3	42	273	1092	3003	6006	9009	10296	9009	6006	3003	1092	273	42	3	49152

b-files: A091629 (row sums), A134059, A134059 (reversed triangle) ?

Technical details

Node id: n-7f71648c3ccd02d4 · revision 1 · JSON
Definition source: oeis_catalog:A134059
Fingerprint: f6-77a7ed3b1d5f257e

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): yes

Matches ?

A134059 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A134059 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A167881, A082505, A042950, A003945, A111286, A007283, A091629, A058764

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 3,  3;
  2: 3,  6,   3;
  3: 3,  9,   9,   3;
  4: 3, 12,  18,  12,   3;
  5: 3, 15,  30,  30,  15,   3;
  6: 3, 18,  45,  60,  45,  18,   3;
  7: 3, 21,  63, 105, 105,  63,  21,   3;
  8: 3, 24,  84, 168, 210, 168,  84,  24,  3;
  9: 3, 27, 108, 252, 378, 378, 252, 108, 27, 3;

%F Formula

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

%t Mathematica

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

A134059 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 3,  3;
  2: 3,  6,   3;
  3: 3,  9,   9,   3;
  4: 3, 12,  18,  12,   3;
  5: 3, 15,  30,  30,  15,   3;
  6: 3, 18,  45,  60,  45,  18,   3;
  7: 3, 21,  63, 105, 105,  63,  21,   3;
  8: 3, 24,  84, 168, 210, 168,  84,  24,  3;
  9: 3, 27, 108, 252, 378, 378, 252, 108, 27, 3;

%F Formula

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

%t Mathematica

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

A134059 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 3,  3;
  2: 3,  6,   3;
  3: 3,  9,   9,   3;
  4: 3, 12,  18,  12,   3;
  5: 3, 15,  30,  30,  15,   3;
  6: 3, 18,  45,  60,  45,  18,   3;
  7: 3, 21,  63, 105, 105,  63,  21,   3;
  8: 3, 24,  84, 168, 210, 168,  84,  24,  3;
  9: 3, 27, 108, 252, 378, 378, 252, 108, 27, 3;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (n - 1)*x)*v[i - 1])/(n - 1)); 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 Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem, Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem, Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
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

Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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.