A156289 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3\,x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 2\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3}\right)\,P_{n-2}''(x) + \left(1 + 6\,x\right)\,P_{n-1}(x) + \left(3\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{2}(x) = 1 + 6\,x\)

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	6														7
3	1	30	45													76
4	1	126	630	420												1177
5	1	510	6615	12600	4725											24451
6	1	2046	63360	263340	259875	62370										650992
7	1	8190	585585	4804800	9459450	5675670	945945									21479641
8	1	32766	5331690	82342260	290540250	329188860	132432300	16216200								856084327
9	1	131070	48230955	1367238600	8189856675	15685930260	11481880410	3308104800	310134825							40391507596
10	1	524286	435061620	22325973180	220127851125	672799597470	800515415700	408550942800	88388425125	6547290750						2219691082057
11	1	2097150	3919486725	361276146000	5754363476400	27126473143770	49429382597595	39870381751200	14972999216175	2520706938750	151242416325					140190747270091
12	1	8388606	35291109150	5817000212100	147923443552500	1052510631064200	2833457923296000	3399065276607000	1975333987501875	568167343994250	76528662660450	3794809718700				10062634378104832
13	1	33554430	317682896895	93401387079000	3763527341198625	39842972173204200	154802516145977700	266113742102208000	225009176402760750	98518229691131250	22384633828181625	2466626317155000	102776096548125			813097919201895601
14	1	134217726	2859397730280	1497387233635020	95138949134604375	1484025559139173050	8189608369113171000	19685036914186644000	23315168606328348750	14602016693171407500	4984311799075108500	916845002086513500	84173623072914375	2988412653476250		73360814174726944327
15	1	536870910	25735586205465	23984883450309600	2395319334743503350	54680754257587007550	423798531066822836925	1400235648835599576000	2265004988096415815250	1952410784339628112500	940408313300347641750	257086594531797030000	39056561105832270000	3033238843278393750	92854250304440625	7338227598581930013676

Technical details

Node id: n-2f3805888228e572 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e828929508d533c3

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
   1: 1;
   2: 1,      6;
   3: 1,     30,        45;
   4: 1,    126,       630,         420;
   5: 1,    510,      6615,       12600,         4725;
   6: 1,   2046,     63360,      263340,       259875,        62370;
   7: 1,   8190,    585585,     4804800,      9459450,      5675670,       945945;
   8: 1,  32766,   5331690,    82342260,    290540250,    329188860,    132432300,     16216200;
   9: 1, 131070,  48230955,  1367238600,   8189856675,  15685930260,  11481880410,   3308104800,   310134825;
  10: 1, 524286, 435061620, 22325973180, 220127851125, 672799597470, 800515415700, 408550942800, 88388425125, 6547290750;

%F Formula

The row polynomials satisfy P(n,x) = -3*x^2*P(n-2,x) - (2*x^2+2*x^3)*d/dx P(n-2,x) - x^3*d^2/dx^2 P(n-2,x) + (1+6*x)*P(n-1,x) + (3*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 6*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-3*x^2*P[n - 2, x] - (2*x^2 + 2*x^3)*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (1 + 6*x)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 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 + 6*x; for(n = 3, N, my(i = n); v[i] = -3*x^2*v[i - 2] - (2*x^2 + 2*x^3)*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (1 + 6*x)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from 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 read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=…?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.