A013618 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(1 + 44\,x + 363\,x^{2} + 11\,n x + 121\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 44\,x + 363\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	44	363								408
1	1	66	1089	5324							6480
2	1	88	2178	21296	73205						96768
3	1	110	3630	53240	366025	966306					1389312
4	1	132	5445	106480	1098075	5797836	12400927				19408896
5	1	154	7623	186340	2562175	20292426	86806489	155897368			265752576
6	1	176	10164	298144	5124350	54113136	347225956	1247178944	1929229929		3583180800
7	1	198	13068	447216	9223830	121754556	1041677868	5612305248	17363069361	23579476910	47727968256

All 15 rows

Technical details

Node id: n-575ba583ca26593b · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3b5c3b5307ad6292

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1,  44,   363;
  1: 1,  66,  1089,   5324;
  2: 1,  88,  2178,  21296,    73205;
  3: 1, 110,  3630,  53240,   366025,    966306;
  4: 1, 132,  5445, 106480,  1098075,   5797836,   12400927;
  5: 1, 154,  7623, 186340,  2562175,  20292426,   86806489,   155897368;
  6: 1, 176, 10164, 298144,  5124350,  54113136,  347225956,  1247178944,   1929229929;
  7: 1, 198, 13068, 447216,  9223830, 121754556, 1041677868,  5612305248,  17363069361,   23579476910;
  8: 1, 220, 16335, 638880, 15373050, 243509112, 2604194670, 18707684160,  86815346805,  235794769100,  285311670611;
  9: 1, 242, 19965, 878460, 24157650, 446433372, 5729228274, 51446131440, 318322938285, 1296871230050, 3138428376721, 3423740047332;

%F Formula

The row polynomials satisfy (1+(11*n+22)*x)*P(n,x) = (1+(11*n+44)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 44*x + 363*x^2.

%t Mathematica

P[0, x_] := 1 + 44*x + 363*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (11*n + 44)*x + (121*n + 363)*x^2)*P[n - 1, x])/(1 + (11*n + 22)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 44*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (11*n + 44)*x + (121*n + 363)*x^2)*v[i - 1])/(1 + (11*n + 22)*x)); 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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of coefficients in expansion of (1+11x)^n.?
  - 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
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

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.