A193729: Mirror of the triangle A193728.

A193729 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A193729

\(\left(-1 + n\right)\,P_{n}(x) = \left(-3 - 4\,x + 3\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	3	10	8						21
3	9	42	64	32					147
4	27	162	360	352	128				1029
5	81	594	1728	2496	1792	512			7203
6	243	2106	7560	14400	15360	8704	2048		50421
7	729	7290	31104	73440	103680	87552	40960	8192	352947

All 15 rows b-files: A169634 (row sums), A193728 (reversed triangle), A193729 ?

Technical details

Node id: n-c6ca397ede50cf7f · revision 1 · JSON
Definition source: oeis_catalog:A193729
Fingerprint: f6-63d3891a7d0f6d93

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 ?

A193729 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A193728 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A169634

Submission text ?

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

%e Example

Triangle begins:
  0:    1;
  1:    1,     2;
  2:    3,    10,      8;
  3:    9,    42,     64,      32;
  4:   27,   162,    360,     352,     128;
  5:   81,   594,   1728,    2496,    1792,     512;
  6:  243,  2106,   7560,   14400,   15360,    8704,    2048;
  7:  729,  7290,  31104,   73440,  103680,   87552,   40960,    8192;
  8: 2187, 24786, 122472,  344736,  604800,  677376,  473088,  188416,  32768;
  9: 6561, 83106, 466560, 1524096, 3193344, 4451328, 4128768, 2457600, 851968, 131072;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (3*n-3+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (4*n - 4)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (4*n - 4)*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

A193728 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:      1;
  1:      2,      1;
  2:      8,     10,       3;
  3:     32,     64,      42,       9;
  4:    128,    352,     360,     162,      27;
  5:    512,   1792,    2496,    1728,     594,      81;
  6:   2048,   8704,   15360,   14400,    7560,    2106,     243;
  7:   8192,  40960,   87552,  103680,   73440,   31104,    7290,    729;
  8:  32768, 188416,  473088,  677376,  604800,  344736,  122472,  24786,  2187;
  9: 131072, 851968, 2457600, 4128768, 4451328, 3193344, 1524096, 466560, 83106, 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+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (4*n - 4)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (4*n - 4)*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

A193729 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    1,     2;
  2:    3,    10,      8;
  3:    9,    42,     64,      32;
  4:   27,   162,    360,     352,     128;
  5:   81,   594,   1728,    2496,    1792,     512;
  6:  243,  2106,   7560,   14400,   15360,    8704,    2048;
  7:  729,  7290,  31104,   73440,  103680,   87552,   40960,    8192;
  8: 2187, 24786, 122472,  344736,  604800,  677376,  473088,  188416,  32768;
  9: 6561, 83106, 466560, 1524096, 3193344, 4451328, 4128768, 2457600, 851968, 131072;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (3*n-3+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (4*n - 4)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (4*n - 4)*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, Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences), 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

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

Used in 5

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.