A136158: Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

A136158 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A136158

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

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

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

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	3	4	1													8
3	9	15	7	1												32
4	27	54	36	10	1											128
5	81	189	162	66	13	1										512
6	243	648	675	360	105	16	1									2048
7	729	2187	2673	1755	675	153	19	1								8192
8	2187	7290	10206	7938	3780	1134	210	22	1							32768
9	6561	24057	37908	34020	19278	7182	1764	276	25	1						131072
10	19683	78732	137781	139968	91854	40824	12474	2592	351	28	1					524288
11	59049	255879	492075	557685	415530	214326	78246	20250	3645	435	31	1				2097152
12	177147	826686	1732104	2165130	1804275	1058508	449064	138996	31185	4950	528	34	1			8388608
13	531441	2657205	6022998	8227494	7577955	4979799	2405700	866052	232551	46035	6534	630	37	1		33554432
14	1594323	8503056	20726199	30705480	30961359	22517352	12196899	5003856	1563705	370656	65637	8424	741	40	1	134217728

b-files: A004171 (row sums), A038763 (reversed triangle), A081294 (row sums), A136158 ?

Technical details

Node id: n-19598058c018a2ec · revision 1 · JSON
Definition source: oeis_catalog:A136158
Fingerprint: f6-fb13869215ab1dd7

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: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A136158 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A038763 reversed triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A004171, A081294, A081654

Submission text ?

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

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    3,     4,     1;
  3:    9,    15,     7,     1;
  4:   27,    54,    36,    10,     1;
  5:   81,   189,   162,    66,    13,    1;
  6:  243,   648,   675,   360,   105,   16,    1;
  7:  729,  2187,  2673,  1755,   675,  153,   19,   1;
  8: 2187,  7290, 10206,  7938,  3780, 1134,  210,  22,  1;
  9: 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1;

%F Formula

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

A038763 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   3;
  3: 1,  7,  15,    9;
  4: 1, 10,  36,   54,   27;
  5: 1, 13,  66,  162,  189,    81;
  6: 1, 16, 105,  360,  675,   648,   243;
  7: 1, 19, 153,  675, 1755,  2673,  2187,   729;
  8: 1, 22, 210, 1134, 3780,  7938, 10206,  7290,  2187;
  9: 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (3*n-3+(n-1)*x)*P(n-1,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[Together[((3*n - 3 + (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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (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

A136158 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    3,     4,     1;
  3:    9,    15,     7,     1;
  4:   27,    54,    36,    10,     1;
  5:   81,   189,   162,    66,    13,    1;
  6:  243,   648,   675,   360,   105,   16,    1;
  7:  729,  2187,  2673,  1755,   675,  153,   19,   1;
  8: 2187,  7290, 10206,  7938,  3780, 1134,  210,  22,  1;
  9: 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1;

%F Formula

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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem, Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem, Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
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

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
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

Related ?

Same fingerprint ?

A038763: Triangular matrix arising in enumeration of catafusenes, read by rows.

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.