A144698 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8\,x^{2} - 6\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} + 4\,x^{3} - n x^{2} + n x^{3}\right)\,P_{n-2}'(x) + \left(1 + 6\,x + 2\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x\)

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	8														9
2	1	26	48													75
3	1	64	339	256												660
4	1	142	1593	3284	1280											6300
5	1	300	6270	25880	26925	6144										65520
6	1	618	22410	162040	328725	198414	28672									740880
7	1	1256	75573	887200	3061475	3557304	1358119	131072								9072000
8	1	2534	245559	4455676	24179375	47060214	34407961	8809256	589824							119750400
9	1	5092	778488	21096824	171023230	514483284	625180808	306414352	54860481	2621440						1696464000
10	1	10210	2427120	95805120	1118826450	4933950588	9178109640	7452761280	2562041205	330994850	11534336					25686460800
11	1	20448	7480455	422054400	6911452650	43029620928	116053248654	143536235520	81748865925	20390223200	1947349371	50331648				414096883200
12	1	40926	22870773	1817878820	40888079250	349408900548	1314384984738	2341980441744	2027565913005	839726744750	155996316201	11225885436	218103808			7083236160000
13	1	81884	69533154	7698436376	233980927075	2685241744488	13684775783532	33757386216672	41960645189703	26420075563100	8181855412786	1155777422184	63642228949	939524096		128152088064000
14	1	163802	210564678	32184408584	1304627271955	19761552929298	133386814907556	442096271012064	757416401644887	683807207528030	322497074495926	76348952218392	8339902220533	355842262454	4026531840	2445351068160000

Technical details

Node id: n-3159219aa7264769 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-261b35285a74aa4e

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,    8;
  2: 1,   26,     48;
  3: 1,   64,    339,      256;
  4: 1,  142,   1593,     3284,      1280;
  5: 1,  300,   6270,    25880,     26925,      6144;
  6: 1,  618,  22410,   162040,    328725,    198414,     28672;
  7: 1, 1256,  75573,   887200,   3061475,   3557304,   1358119,    131072;
  8: 1, 2534, 245559,  4455676,  24179375,  47060214,  34407961,   8809256,   589824;
  9: 1, 5092, 778488, 21096824, 171023230, 514483284, 625180808, 306414352, 54860481, 2621440;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+6*n+8)*x^2*P(n-2,x) - ((n+4)*x^2-(n+4)*x^3)*d/dx P(n-2,x) + (1+(2*n+6)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 8)*x^2*P[n - 2, x] - ((n + 4)*x^2 - (n + 4)*x^3)*D[P[n - 2, x], x] + (1 + (2*n + 6)*x)*P[n - 1, x] + (x - 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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 8)*x^2*v[i - 2] - ((n + 4)*x^2 - (n + 4)*x^3)*deriv(v[i - 2], x) + (1 + (2*n + 6)*x)*v[i - 1] + (x - 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 of 4-Eulerian numbers.?
  - 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.