A193727: Mirror of the triangle A193726.

A193727 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A193727

\(\left(-1 + n\right)\,P_{n}(x) = \left(-5 - 2\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

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	10	9	2													21
3	50	65	28	4												147
4	250	425	270	76	8											1029
5	1250	2625	2200	920	192	16										7203
6	6250	15625	16250	9000	2800	464	32									50421
7	31250	90625	112500	77500	32000	7920	1088	64								352947
8	156250	515625	743750	612500	315000	103600	21280	2496	128							2470629
9	781250	2890625	4750000	4550000	2800000	1148000	313600	55040	5632	256						17294403
10	3906250	16015625	29531250	32250000	23100000	11340000	3864000	902400	138240	12544	512					121060821
11	19531250	87890625	179687500	220312500	180000000	102900000	42000000	12240000	2496000	339200	27648	1024				847425747
12	97656250	478515625	1074218750	1460937500	1340625000	874500000	415800000	145200000	36960000	6688000	816640	60416	2048			5931980229
13	488281250	2587890625	6328125000	9453125000	9625000000	7053750000	3828000000	1557600000	475200000	107360000	17459200	1935360	131072	4096		41523861603
14	2441406250	13916015625	36816406250	59921875000	67031250000	54518750000	33247500000	15444000000	5491200000	1487200000	302016000	44595200	4526080	282624	8192	290667031221

b-files: A169634 (row sums), A193726 (reversed triangle), A193727 ?

Technical details

Node id: n-2ed1083d14adf57b · revision 1 · JSON
Definition source: oeis_catalog:A193727
Fingerprint: f6-ae9eedd6404e58f4

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 ?

A193726 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A193727 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:      2,       1;
  2:     10,       9,       2;
  3:     50,      65,      28,       4;
  4:    250,     425,     270,      76,       8;
  5:   1250,    2625,    2200,     920,     192,      16;
  6:   6250,   15625,   16250,    9000,    2800,     464,     32;
  7:  31250,   90625,  112500,   77500,   32000,    7920,   1088,    64;
  8: 156250,  515625,  743750,  612500,  315000,  103600,  21280,  2496,  128;
  9: 781250, 2890625, 4750000, 4550000, 2800000, 1148000, 313600, 55040, 5632, 256;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n-5+(2*n-2)*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[((5*n - 5 + (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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n - 5 + (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

A193726 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:   1;
  1:   1,    2;
  2:   2,    9,    10;
  3:   4,   28,    65,     50;
  4:   8,   76,   270,    425,     250;
  5:  16,  192,   920,   2200,    2625,    1250;
  6:  32,  464,  2800,   9000,   16250,   15625,    6250;
  7:  64, 1088,  7920,  32000,   77500,  112500,   90625,   31250;
  8: 128, 2496, 21280, 103600,  315000,  612500,  743750,  515625,  156250;
  9: 256, 5632, 55040, 313600, 1148000, 2800000, 4550000, 4750000, 2890625, 781250;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (5*n-5+(2*n-2)*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[((5*n - 5 + (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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n - 5 + (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

A193727 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:      1;
  1:      2,       1;
  2:     10,       9,       2;
  3:     50,      65,      28,       4;
  4:    250,     425,     270,      76,       8;
  5:   1250,    2625,    2200,     920,     192,      16;
  6:   6250,   15625,   16250,    9000,    2800,     464,     32;
  7:  31250,   90625,  112500,   77500,   32000,    7920,   1088,    64;
  8: 156250,  515625,  743750,  612500,  315000,  103600,  21280,  2496,  128;
  9: 781250, 2890625, 4750000, 4550000, 2800000, 1148000, 313600, 55040, 5632, 256;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n-5+(2*n-2)*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[((5*n - 5 + (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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n - 5 + (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 Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem
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

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
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Used in 5

Related ?

Same fingerprint ?

A193726: Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(x+2)^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.