A357613 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 8\,n - 10\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(-6 - 6\,x + 33\,n + 35\,n x + 2\,n x^{2} - 54\,n^{2} - 55\,n^{2} x - 3\,n^{2} x^{2} + 27\,n^{3} + 24\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2} + 9\,n x + 10\,n x^{2} + n x^{3} - 9\,n^{2} x - 10\,n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	20	12													32
3	168	210	60												438
4	1320	2520	1512	280											5632
5	10010	25740	23760	9240	1260										70010
6	74256	240240	300300	180180	51480	5544									852000
7	542640	2116296	3341520	2722720	1201200	270270	24024								10218670
8	3922512	17907120	34186320	35271600	21162960	7351344	1361360	102960							121266176
9	28120950	147094200	329491008	411863760	313374600	148140720	42325920	6651216	437580						1427499954
10	200300100	1181079900	3037062600	4461857400	4118637600	2471182560	961015440	232792560	31744440	1847560					16697520160
11	1419269280	9313954650	27040513500	45668422800	49605355800	36141044940	17847429600	5883768000	1235591280	148728580	7759752				194311838182
12	10013421600	72382733280	234179431200	447069823200	558837279000	479518439400	287711063640	120470149800	34420042800	6374082000	686439600	32449872			2251695355392
13	70394353848	555744898800	1982657476800	4222326108000	5971575495600	5901321666240	4172651683200	2123581660200	770654634750	194090796900	32125373280	3129094800	135207800		26000388450218
14	493362138080	4223661230880	16472278800432	38716894616400	61131938868000	68401682949600	55734704625600	33440822775360	14753304165600	4719070356000	1061790830100	158801561100	14115694320	561632400	299322990243872

Technical details

Node id: n-0da20f282347d5c3 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-69f4d9beddd80e10

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        0;
  1:        2;
  2:       20,        12;
  3:      168,       210,        60;
  4:     1320,      2520,      1512,       280;
  5:    10010,     25740,     23760,      9240,      1260;
  6:    74256,    240240,    300300,    180180,     51480,      5544;
  7:   542640,   2116296,   3341520,   2722720,   1201200,    270270,    24024;
  8:  3922512,  17907120,  34186320,  35271600,  21162960,   7351344,  1361360,  102960;
  9: 28120950, 147094200, 329491008, 411863760, 313374600, 148140720, 42325920, 6651216, 437580;

%F Formula

The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (27*n^3-54*n^2+33*n-6+(24*n^3-55*n^2+35*n-6)*x+(n^3-3*n^2+2*n)*x^2)*P(n-1,x) - ((9*n^2-9*n+2)*x+(10*n^2-10*n+2)*x^2+(n^2-n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 55*n^2 + 35*n - 6)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 10*n + 2)*x^2 + (n^2 - n)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 10*n^2 + 8*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 55*n^2 + 35*n - 6)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 1] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 10*n + 2)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 10*n^2 + 8*n - 2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*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

References

[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.