A280580 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n - 12\,n^{2} + n^{2} x - 3\,n^{3} x + 2\,n^{4} x\right)\,P_{n}(x) = \left(-36 - 72\,x - 6\,x^{2} + 96\,n + 138\,n x + 13\,n x^{2} - 48\,n^{2} - 50\,n^{2} x - 3\,n^{2} x^{2} - 24\,n^{3} x - 10\,n^{3} x^{2} + 8\,n^{4} x + 6\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(72\,x - 6\,x^{2} - 3\,x^{3} - 96\,n x + 4\,n x^{2} + 5\,n x^{3} - 4\,n^{2} x^{2} + n^{2} x^{3} + 16\,n^{3} x^{2} - 4\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

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	2														3
3	2	12	3													17
4	5	60	45	4												114
5	14	280	420	112	5											831
6	42	1260	3150	1680	225	6										6363
7	132	5544	20790	18480	4950	396	7									50299
8	429	24024	126126	168168	75075	12012	637	8								406479
9	1430	102960	720720	1345344	900900	240240	25480	960	9							3338043
10	4862	437580	3938220	9801792	9189180	3675672	649740	48960	1377	10						27747393
11	16796	1847560	20785050	66512160	83140200	46558512	12345060	1550400	87210	1900	11					232844859
12	58786	7759752	106696590	426786360	685906650	512143632	190113924	35814240	3357585	146300	2541	12				1968786372
13	208012	32449872	535422888	2617623008	5258617650	5048272944	2498640144	658982016	92669346	6729800	233772	3312	13			16749852777
14	742900	135207800	2636552100	15467772320	37978905250	45574686300	29002073100	10198531200	2007835830	218718500	12662650	358800	4225	14		143234050989
15	2674440	561632400	12777137100	88588150560	261018657900	382827364920	304521767550	137680171200	36141044940	5511706200	478648170	22604400	532350	5292	15	1230132097437

Technical details

Node id: n-d0d869eaaa277eca · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4f57bdcd9ba55419

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,      2;
   3:    2,     12,       3;
   4:    5,     60,      45,       4;
   5:   14,    280,     420,     112,       5;
   6:   42,   1260,    3150,    1680,     225,       6;
   7:  132,   5544,   20790,   18480,    4950,     396,      7;
   8:  429,  24024,  126126,  168168,   75075,   12012,    637,     8;
   9: 1430, 102960,  720720, 1345344,  900900,  240240,  25480,   960,    9;
  10: 4862, 437580, 3938220, 9801792, 9189180, 3675672, 649740, 48960, 1377, 10;

%F Formula

The row polynomials satisfy -(12*n^2-6*n-(2*n^4-3*n^3+n^2)*x)*P(n,x) = -(48*n^2-96*n+36-(8*n^4-24*n^3-50*n^2+138*n-72)*x-(6*n^4-10*n^3-3*n^2+13*n-6)*x^2)*P(n-1,x) - ((96*n-72)*x-(16*n^3-4*n^2+4*n-6)*x^2+(4*n^3-n^2-5*n+3)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n^2 - 96*n + 36 - (8*n^4 - 24*n^3 - 50*n^2 + 138*n - 72)*x - (6*n^4 - 10*n^3 - 3*n^2 + 13*n - 6)*x^2)*P[n - 1, x] - ((96*n - 72)*x - (16*n^3 - 4*n^2 + 4*n - 6)*x^2 + (4*n^3 - n^2 - 5*n + 3)*x^3)*D[P[n - 1, x], x])/(-12*n^2 + 6*n + (2*n^4 - 3*n^3 + n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(48*n^2 - 96*n + 36 - (8*n^4 - 24*n^3 - 50*n^2 + 138*n - 72)*x - (6*n^4 - 10*n^3 - 3*n^2 + 13*n - 6)*x^2)*v[i - 1] - ((96*n - 72)*x - (16*n^3 - 4*n^2 + 4*n - 6)*x^2 + (4*n^3 - n^2 - 5*n + 3)*x^3)*deriv(v[i - 1], x))/(-12*n^2 + 6*n + (2*n^4 - 3*n^3 + n^2)*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
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 read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=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

References

[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.