A200139: Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

A200139 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A200139

\(\left(-2 + n\right)\,P_{n}(x) = \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	2	3	1													6
4	4	8	5	1												18
5	8	20	18	7	1											54
6	16	48	56	32	9	1										162
7	32	112	160	120	50	11	1									486
8	64	256	432	400	220	72	13	1								1458
9	128	576	1120	1232	840	364	98	15	1							4374
10	256	1280	2816	3584	2912	1568	560	128	17	1						13122
11	512	2816	6912	9984	9408	6048	2688	816	162	19	1					39366
12	1024	6144	16640	26880	28800	21504	11424	4320	1140	200	21	1				118098
13	2048	13312	39424	70400	84480	71808	44352	20064	6600	1540	242	23	1			354294
14	4096	28672	92160	180224	239360	228096	160512	84480	33264	9680	2024	288	25	1		1062882
15	8192	61440	212992	452608	658944	695552	549120	329472	151008	52624	13728	2600	338	27	1	3188646

b-files: A008776 (row sums), A025192 (row sums), A081277 (reversed triangle), A200139 ?

Technical details

Node id: n-f2a2a3d7ea29bcbf · revision 1 · JSON
Definition source: oeis_catalog:A200139
Fingerprint: f6-04668acf2e74f66d

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 -1)
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 ?

A081277 reversed triangle 64 terms, 10 rows · tabl · nonn b-file?
A200139 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A110593, A099856, A025192, A027334, A008776

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   1,    1;
   3:   2,    3,    1;
   4:   4,    8,    5,    1;
   5:   8,   20,   18,    7,    1;
   6:  16,   48,   56,   32,    9,    1;
   7:  32,  112,  160,  120,   50,   11,   1;
   8:  64,  256,  432,  400,  220,   72,  13,   1;
   9: 128,  576, 1120, 1232,  840,  364,  98,  15,  1;
  10: 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A081277 reversed triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   2;
  3: 1,  5,   8,   4;
  4: 1,  7,  18,  20,    8;
  5: 1,  9,  32,  56,   48,   16;
  6: 1, 11,  50, 120,  160,  112,   32;
  7: 1, 13,  72, 220,  400,  432,  256,   64;
  8: 1, 15,  98, 364,  840, 1232, 1120,  576,  128;
  9: 1, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256;

%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+(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[((2*n - 2 + (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] = ((2*n - 2 + (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

A200139 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:   1;
  1:   1,    1;
  2:   2,    3,    1;
  3:   4,    8,    5,    1;
  4:   8,   20,   18,    7,    1;
  5:  16,   48,   56,   32,    9,    1;
  6:  32,  112,  160,  120,   50,   11,   1;
  7:  64,  256,  432,  400,  220,   72,  13,   1;
  8: 128,  576, 1120, 1232,  840,  364,  98,  15,  1;
  9: 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(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[((2*n - 2 + (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] = ((2*n - 2 + (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 ≥ 1, 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 ≥ 1: 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 ≥ 1, 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 ≥ 3
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 2
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

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.