A008955 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x + 16\,n^{3} x^{2} - 8\,n^{4} - 28\,n^{4} x - 24\,n^{4} x^{2} + 4\,n^{5} + 16\,n^{5} x + 16\,n^{5} x^{2} - n^{6} - 4\,n^{6} x - 4\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 4\,n^{2} x - 4\,n^{3} - 8\,n^{3} x + 2\,n^{4} + 4\,n^{4} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	10	52	48													110
3	100	880	1884	1152												4016
4	1700	21360	74268	99968	46080											243376
5	44200	725360	3532968	7550368	7446080	2764800										22063776
6	1635400	33203120	209058696	618528544	955038080	745638912	232243200									2795345952
7	81770000	1980694400	15333793440	58243430144	123521650640	149594423808	96863542272	26011238400								471630543104
8	5315050000	149678256000	1376989898400	6402790373120	17347856114640	28696563085824	28635564203008	15860482965504	3745618329600							102220858276096
9	435834100000	13995693192000	149284987876800	822458628650240	2719089251957280	5725541401723584	7771766687867392	6602229775613952	3197713723490304	674211299328000						27676727283799552
10	44019244100000	1587898652392000	19276491733156800	122877651594154240	480242671610245280	1230861102043829184	2120908095876776192	2443229021706698752	1808470785585659904	778698390896640000	148326485852160000					9154522614795812352
11	5370347780200000	215028949736224000	2928139202263425600	21210954827052078080	95760095543681581760	289627526284962389760	606263905507341135040	884656522593565634560	885802537001099243520	581278903813534187520	225385342930649088000	39158192264970240000				3632292519266635427840
12	778700428129000000	34272518033147680000	517472690614245480000	4199993903591706752000	21521157591572577464000	75090680331215901171456	185223115130311827540160	327822629817340442030080	415070081185474655907840	367742252893314215116800	216830031453071748366336	76489914711904616448000	12217355986670714880000			1702759736913543927836672
13	132379072781930000000	6352729555050309600000	105346524047227605360000	947206656180743444160000	5433094214834834271600000	21494397175448540618545920	61098687849429147710438016	127278890399285472526341120	195216468789574394586270720	218398057925963898455244800	173587674972756422402703360	92954431644083694921056256	30085007621409761918976000	4447117579148140216320000		931052866460790110916616192
14	26078677338040210000000	1355272915405944111200000	24488670215673420300720000	241660827836290753567680000	1534450821851026639143600000	6790123762221668543214186240	21866545814576008008504623232	52446153565203496261965431808	94587635017631049117611765760	128052034928651735415481139200	128370014547308648227234119680	92544425131325052534204137472	45401395139605265541180162048	13576583842038089269641216000	1867789383242218890854400000	587304682803298966607224381440

Technical details

Node id: n-624a28504ef54c1f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-66ed502267415f23

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,              4;
  2:           10,             52,              48;
  3:          100,            880,            1884,            1152;
  4:         1700,          21360,           74268,           99968,            46080;
  5:        44200,         725360,         3532968,         7550368,          7446080,          2764800;
  6:      1635400,       33203120,       209058696,       618528544,        955038080,        745638912,        232243200;
  7:     81770000,     1980694400,     15333793440,     58243430144,     123521650640,     149594423808,      96863542272,      26011238400;
  8:   5315050000,   149678256000,   1376989898400,   6402790373120,   17347856114640,   28696563085824,   28635564203008,   15860482965504,    3745618329600;
  9: 435834100000, 13995693192000, 149284987876800, 822458628650240, 2719089251957280, 5725541401723584, 7771766687867392, 6602229775613952, 3197713723490304, 674211299328000;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(4*n^6-16*n^5+28*n^4-24*n^3+8*n^2)*x+(4*n^6-16*n^5+24*n^4-16*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(4*n^4-8*n^3+4*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*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 basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) 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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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.