A290319 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-49 + 28\,x + 9\,x^{2} + 56\,n - 16\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(-28\,x^{2} + 4\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(-10 + 4\,x + 8\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	5	24	9													38
3	45	236	135	16												432
4	585	3248	2286	448	25											6592
5	9945	57556	46170	11680	1125	36										126512
6	208845	1248456	1099071	327360	41875	2376	49									2928032
7	5221125	32046780	30285801	10137904	1558375	119700	4459	64								79374208
8	151412625	950241120	950393484	347840640	61033350	5715360	292236	7680	81							2466936576
9	4996616625	31963607460	33501027492	13168329536	2557601550	276494904	17423028	635136	12393	100						86481748224
10	184874815125	1202639942520	1311456133989	546785575040	115206772250	13913257680	1020992322	46256640	1262385	19000	121					3375945027072
11	7579867420125	50047736903820	56475641364219	24749686148176	5577830123250	736341316920	60798174822	3230063232	110301345	2337500	27951	144				145231244181504
12	341094033905625	2282467630352400	2654011269423450	1214137016870976	289673740152775	41167434638880	3738160215020	224762706432	9051609303	241362000	4086170	39744	169			6826523345362944
13	16713607661375625	113205290022890100	135182104370041950	64210956083430624	16091102356346875	2434334483125116	239203303238900	15895867589888	727994156175	23001564300	492270350	6810336	54925	196		348093245642895360
14	885821206052908125	6066734801858677800	7419363434163726075	3643504413524119872	953158043766744725	152190914998770648	15991174784804219	1154909786494464	58701897695727	2117841125400	53922221353	946790208	10903711	74088	225	19137979773555056640

Technical details

Node id: n-c1ad8f9a14dca9fa · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e37871200d1c6792

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:          1;
  1:          1,           4;
  2:          5,          24,           9;
  3:         45,         236,         135,          16;
  4:        585,        3248,        2286,         448,         25;
  5:       9945,       57556,       46170,       11680,       1125,        36;
  6:     208845,     1248456,     1099071,      327360,      41875,      2376,       49;
  7:    5221125,    32046780,    30285801,    10137904,    1558375,    119700,     4459,     64;
  8:  151412625,   950241120,   950393484,   347840640,   61033350,   5715360,   292236,   7680,    81;
  9: 4996616625, 31963607460, 33501027492, 13168329536, 2557601550, 276494904, 17423028, 635136, 12393, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^2-56*n+49+(16*n-28)*x-9*x^2)*P(n-2,x) + ((16*n-28)*x^2+4*x^3)*d/dx P(n-2,x) + (8*n-10+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 56*n + 49 + (16*n - 28)*x - 9*x^2)*P[n - 2, x] + ((16*n - 28)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (8*n - 10 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 56*n + 49 + (16*n - 28)*x - 9*x^2)*v[i - 2] + ((16*n - 28)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (8*n - 10 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], 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 read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
  - 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.