A008955 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 12\,n^{2} x - 9\,n^{2} x^{2} + 8\,n^{3} + 36\,n^{3} x + 36\,n^{3} x^{2} - 8\,n^{4} - 42\,n^{4} x - 54\,n^{4} x^{2} + 4\,n^{5} + 24\,n^{5} x + 36\,n^{5} x^{2} - n^{6} - 6\,n^{6} x - 9\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 6\,n^{2} x - 4\,n^{3} - 12\,n^{3} x + 2\,n^{4} + 6\,n^{4} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	39	72													111
3	660	2826	2916												6402
4	16020	111402	253044	186624											567090
5	544020	5299452	19111869	30156624	17496000										72607965
6	24902340	313588044	1565650377	3867904224	4718496240	2267481600									12758022825
7	1485520800	23000690160	147428682552	500262685092	946652213160	945716816736	388873094400								2953419702900
8	112258692000	2065484847600	16207063131960	70258817264292	181595438277480	279580262999904	237117318084864	85329867571200							872266510869300
9	10496769894000	223927481815200	2081848403770920	11012311470426984	36231941682782055	75878811367705296	98704624643431632	72847915763263488	23327052547276800						320318930130366375
10	1190923989294000	28914737599735200	311034055597702920	1944982820021493384	7789042911371106555	20707258953939774696	36526751067683448432	41199225084123314688	26942203877376384000	7775684182425600000					143226288614127853875
11	161271712302168000	4392208803395138400	53690229405975572640	387828386951910406128	1832799189772027622700	5919192684573906528270	13225787797250875292460	20179689046056292141440	20111682416768777483712	11815322367464398080000	3104830694042542080000				76635376292802402513750
12	25704388524860760000	776209035921368220000	10631234568466507716000	87160688245868938729200	475183211470975624600620	1808406039241035566206830	4901012343626625807654780	9455815287006594504900480	12723522826814830715878080	11366829303331350913781760	6064841639655250569216000	1463222028901503467520000			48357426516286948845183750
13	4764547166287732200000	158019786070841408040000	2397616848457506843030000	22004031570081078799980000	136019232125885296101735900	596530492529917616262178308	1902844270627598924712652545	4447275179612491676668479840	7556359524884907562431884880	9099945507338684291954217600	7370303781209287001382807552	3603156520882577813397504000	803674699374150779535360000		35540673641337276677230070625
14	1016454686554458083400000	36733005323510130451080000	611703970460610969968190000	6214525828496657888531580000	42968751932808996250027272300	213491677216418435333033977716	784080239189160472924246870653	2154824560245407337710593038720	4430475357715990033915280735520	6729510821203860412970478988800	7337794599044890539308401031424	5437536696451388654853589204992	2453534619578909770702675968000	508911548096000709010391040000	30099992849933802696425752378125

Technical details

Node id: n-cf804873205ca95a · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-123cfe6eac72e026

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:              0;
  1:              3;
  2:             39,              72;
  3:            660,            2826,             2916;
  4:          16020,          111402,           253044,            186624;
  5:         544020,         5299452,         19111869,          30156624,          17496000;
  6:       24902340,       313588044,       1565650377,        3867904224,        4718496240,        2267481600;
  7:     1485520800,     23000690160,     147428682552,      500262685092,      946652213160,      945716816736,      388873094400;
  8:   112258692000,   2065484847600,   16207063131960,    70258817264292,   181595438277480,   279580262999904,   237117318084864,    85329867571200;
  9: 10496769894000, 223927481815200, 2081848403770920, 11012311470426984, 36231941682782055, 75878811367705296, 98704624643431632, 72847915763263488, 23327052547276800;

%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+(6*n^6-24*n^5+42*n^4-36*n^3+12*n^2)*x+(9*n^6-36*n^5+54*n^4-36*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(6*n^4-12*n^3+6*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*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] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*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 basis_map_A013610 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.