A144696 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

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	1	14	12													27
3	1	36	99	32												168
4	1	82	513	524	80											1200
5	1	176	2154	4832	2365	192										9720
6	1	366	8046	33688	35985	9666	448									88200
7	1	748	27981	199120	390475	230724	36967	1024								887040
8	1	1514	92919	1057636	3445775	3727194	1335481	134936	2304							9797760
9	1	3048	299196	5221680	26483190	47172744	31090332	7184544	476145	5120						117936000
10	1	6118	943308	24477072	184632750	505495260	547841532	235135392	36615753	1638350	11264					1536796800
11	1	12260	2930871	110485728	1198530810	4805445960	7963102014	5637710784	1652178357	179120180	5530459	24576				21555072000
12	1	24546	9013293	484929020	7373725650	41776808508	100592543058	108806502384	52924153005	10969970450	848963841	18394596	53248			323805081600
13	1	49120	27518526	2083928768	43536402275	339145558848	1142804290740	1790155269888	1333168583319	462527391200	69673439726	3925304640	60474661	114688		5187108326400
14	1	98270	83587098	8812702856	248940942895	2609553863118	11956328221740	26075800578144	28109914678863	14994362893370	3819088618906	427140599640	17795824033	196985306	245760	88268019840000

Technical details

Node id: n-d1639996a4cbf0ae · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-2026a779b361834d

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: 1,   14,     12;
  3: 1,   36,     99,      32;
  4: 1,   82,    513,     524,       80;
  5: 1,  176,   2154,    4832,     2365,      192;
  6: 1,  366,   8046,   33688,    35985,     9666,      448;
  7: 1,  748,  27981,  199120,   390475,   230724,    36967,    1024;
  8: 1, 1514,  92919, 1057636,  3445775,  3727194,  1335481,  134936,   2304;
  9: 1, 3048, 299196, 5221680, 26483190, 47172744, 31090332, 7184544, 476145, 5120;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+2*n)*x^2*P(n-2,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-2,x) + (1+(2*n+2)*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 + 4*x.

%t Mathematica

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