A008955 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-n^{2} - 2\,n^{2} x - n^{2} x^{2} + 4\,n^{3} x + 4\,n^{3} x^{2} - 2\,n^{4} x - 6\,n^{4} x^{2} + 4\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 2\,n^{2} + 2\,n^{2} x - 4\,n^{3} x + 2\,n^{4} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	6	13	4													23
3	15	112	157	36												320
4	31	576	2733	3124	576											7040
5	56	2101	23958	91949	93076	14400										225540
6	92	6097	136422	1229657	4161976	3883536	518400									9936180
7	141	15064	582322	10093512	78933673	245704024	216213264	25401600								576943600
8	205	33048	2023986	59625800	883348713	6276124056	18399002896	15488752896	1625702400							42734614000
9	286	66177	6031056	277772894	6906909906	91898844681	609147480376	1706941816848	1387896581376	131681894400						3934757398000
10	386	123277	15943356	1080815494	41578537406	919349165781	11307886138076	71108246118448	192358031443776	152089529472000	13168189440000					440993907198000
11	507	216568	38294839	3650531940	204891738281	6950045891912	140922727757493	1632202558846684	9847863023064880	25896735177880896	20009352177792000	1593350922240000				59127585214256000
12	651	362440	85037095	10997955540	861529245785	42329146085240	1307551322618565	24800368328876140	273905265934608880	1598330348375192640	4106618070566529024	3112382597326848000	229442532802560000			9346829936415920000
13	820	582309	176860710	30149457195	3183652295460	216954583417755	9648610126625910	277174933338118725	4985150495526346860	52973876620317118960	300727674200615538240	757814038947715885056	564997889524672512000	38775788043632640000		1720561461544507400000
14	1016	903553	347979294	76350060735	10567385332140	965454715424655	59238140821136430	2437454111367375345	66068579347430203440	1137878079877429036640	11711114002927591592704	64965448876102527267072	161035427934936831614976	118378416591431442432000	7600054456551997440000	364896906189526237800000

Technical details

Node id: n-b1d3983f7d4ec8d1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-612ba72480ba5cfa

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:   2,     1;
  2:   6,    13,       4;
  3:  15,   112,     157,        36;
  4:  31,   576,    2733,      3124,        576;
  5:  56,  2101,   23958,     91949,      93076,       14400;
  6:  92,  6097,  136422,   1229657,    4161976,     3883536,       518400;
  7: 141, 15064,  582322,  10093512,   78933673,   245704024,    216213264,      25401600;
  8: 205, 33048, 2023986,  59625800,  883348713,  6276124056,  18399002896,   15488752896,    1625702400;
  9: 286, 66177, 6031056, 277772894, 6906909906, 91898844681, 609147480376, 1706941816848, 1387896581376, 131681894400;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(2*n^4-4*n^3+2*n^2)*x+(n^6-4*n^5+6*n^4-4*n^3+n^2)*x^2)*P(n-2,x) + (2*n^2-2*n+1+(2*n^4-4*n^3+2*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (2*n^4 - 4*n^3 + 2*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*P[n - 2, x] + (2*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (2*n^4 - 4*n^3 + 2*n^2)*x + (n^6 - 4*n^5 + 6*n^4 - 4*n^3 + n^2)*x^2)*v[i - 2] + (2*n^2 - 2*n + 1 + (2*n^4 - 4*n^3 + 2*n^2)*x)*v[i - 1])/(n^2 - 2*n + 1)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - 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.