A193731: Mirror of the triangle A193730.

A193731 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A193731

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	3	8	4													15
3	9	30	28	8												75
4	27	108	144	80	16											375
5	81	378	648	528	208	32										1875
6	243	1296	2700	2880	1680	512	64									9375
7	729	4374	10692	14040	10800	4896	1216	128								46875
8	2187	14580	40824	63504	60480	36288	13440	2816	256							234375
9	6561	48114	151632	272160	308448	229824	112896	35328	6400	512						1171875
10	19683	157464	551124	1119744	1469664	1306368	798336	331776	89856	14336	1024					5859375
11	59049	511758	1968300	4461480	6648480	6858432	5007744	2592000	933120	222720	31744	2048				29296875
12	177147	1653372	6928416	17321040	28868400	33872256	28740096	17791488	7983360	2534400	540672	69632	4096			146484375
13	531441	5314410	24091992	65819952	121247280	159353568	153964800	110854656	59533056	23569920	6690816	1290240	151552	8192		732421875
14	1594323	17006112	82904796	245643840	495381744	720555264	780601536	640493568	400308480	189775872	67212288	17252352	3035136	327680	16384	3662109375

b-files: A005053 (row sums), A193730 (reversed triangle), A193731 ?

Technical details

Node id: n-fd57fb52999582af · revision 1 · JSON
Definition source: oeis_catalog:A193731
Fingerprint: f6-33d8e4d36deefb8f

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 ?

A193730 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A193731 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A005053

Submission text ?

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

%e Example

Triangle begins:
  0:    1;
  1:    1,     2;
  2:    3,     8,      4;
  3:    9,    30,     28,      8;
  4:   27,   108,    144,     80,     16;
  5:   81,   378,    648,    528,    208,     32;
  6:  243,  1296,   2700,   2880,   1680,    512,     64;
  7:  729,  4374,  10692,  14040,  10800,   4896,   1216,   128;
  8: 2187, 14580,  40824,  63504,  60480,  36288,  13440,  2816,  256;
  9: 6561, 48114, 151632, 272160, 308448, 229824, 112896, 35328, 6400, 512;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (3*n-3+(2*n-2)*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 + (2*n - 2)*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 + (2*n - 2)*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

A193730 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:   1;
  1:   2,    1;
  2:   4,    8,     3;
  3:   8,   28,    30,      9;
  4:  16,   80,   144,    108,     27;
  5:  32,  208,   528,    648,    378,     81;
  6:  64,  512,  1680,   2880,   2700,   1296,    243;
  7: 128, 1216,  4896,  10800,  14040,  10692,   4374,    729;
  8: 256, 2816, 13440,  36288,  60480,  63504,  40824,  14580,  2187;
  9: 512, 6400, 35328, 112896, 229824, 308448, 272160, 151632, 48114, 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+(2*n-2)*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 + (2*n - 2)*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 + (2*n - 2)*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

A193731 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    1,     2;
  2:    3,     8,      4;
  3:    9,    30,     28,      8;
  4:   27,   108,    144,     80,     16;
  5:   81,   378,    648,    528,    208,     32;
  6:  243,  1296,   2700,   2880,   1680,    512,     64;
  7:  729,  4374,  10692,  14040,  10800,   4896,   1216,   128;
  8: 2187, 14580,  40824,  63504,  60480,  36288,  13440,  2816,  256;
  9: 6561, 48114, 151632, 272160, 308448, 229824, 112896, 35328, 6400, 512;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (3*n-3+(2*n-2)*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 + (2*n - 2)*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 + (2*n - 2)*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 ?

A193730: Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

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.