A008955 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	10	12													23
3	1	28	147	144												320
4	1	60	819	3280	2880											7040
5	1	110	3069	30580	105380	86400										225540
6	1	182	9009	177892	1481480	4638816	3628800									9936180
7	1	280	22386	766480	12377365	91749840	268814448	203212800								576943600
8	1	408	49266	2676752	73695765	1042331472	7119469168	19865069568	14631321600							42734614000
9	1	570	98838	7997480	344716905	8205559830	105619793272	678924501120	1824835785984	1316818944000						3934757398000
10	1	770	184338	21175880	1344401905	49571588430	1062935106772	12749758017920	78203842161984	204076350720000	144850083840000					440993907198000
11	1	1012	324093	50915744	4547253755	244778745036	8060791006807	159738498497248	1813764652351344	10718148463608960	27307412364672000	19120211066880000				59127585214256000
12	1	1300	542685	113141600	13712087675	1030544193900	49183620172855	1486314389903200	27691401408905520	300920492839824000	1725062129000331264	4308866444353536000	2982752926433280000			9346829936415920000
13	1	1638	872235	235426620	37613250675	3811355574390	252372583736805	10985779314717480	310276924789251420	5500750312956527200	57666181747923612864	322347593500050972672	791864384446826496000	542861032610856960000		1720561461544507400000
14	1	2030	1353807	463370700	95292772575	12657992133150	1123902558413985	67517238071761800	2732641263684455760	73072169489282392000	1243627949221350877184	12652425727237899832320	69237003404290983026688	167687158795848695808000	114000816848279961600000	364896906189526237800000

Technical details

Node id: n-408fa499fa8b44f9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-85270b73446d424a

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,   2;
  2: 1,  10,    12;
  3: 1,  28,   147,     144;
  4: 1,  60,   819,    3280,      2880;
  5: 1, 110,  3069,   30580,    105380,      86400;
  6: 1, 182,  9009,  177892,   1481480,    4638816,      3628800;
  7: 1, 280, 22386,  766480,  12377365,   91749840,    268814448,    203212800;
  8: 1, 408, 49266, 2676752,  73695765, 1042331472,   7119469168,  19865069568,   14631321600;
  9: 1, 570, 98838, 7997480, 344716905, 8205559830, 105619793272, 678924501120, 1824835785984, 1316818944000;

%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) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 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] = 1 + 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 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 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.