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.

A193730 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A193730

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

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

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

Rows ?

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

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

Technical details

Node id: n-5406e3c8f7504399 · revision 1 · JSON
Definition source: oeis_catalog:A193730
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 ?

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

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

%t Mathematica

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

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

%t Mathematica

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

A193731 reversed 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

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

%t Mathematica

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

Related ?

Same fingerprint ?

A193731: Mirror of the triangle A193730.

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.