A008955 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 5 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	3														8
2	49	111	36													196
3	760	3486	4329	972												9547
4	17720	127422	308745	270972	46656											771515
5	588220	5843472	21761595	36527247	25035156	3499200										93254890
6	26537740	338490384	1722444399	4389787683	5685472296	3211180848	377913600									15751826950
7	1567290800	24486210960	158929027632	549405579276	1071717884433	1135047259368	546492563856	55553299200								3543199115525
8	117573742000	2177743539600	17239805555760	75661171641612	199160142593553	315899350655400	283714028584848	119203770154752	10666233446400							1023839819913925
9	10932603994000	234424251709200	2193812144678520	11706260938350624	38985019550388801	83125199704261707	111351093204715848	86948576426610864	32433042017684736	2591894727475200						369580255569869500
10	1234943233394000	30105661589029200	325491424397570520	2048660838554061024	8275288616376479901	22265067536213996007	39977960893340077548	46417332379506664464	32092107012891798336	10769262391022976000	777568418242560000					162980080115368607000
11	166642060082368000	4553480515697306400	55886333807673141840	405725130087235597008	1929756286510005224232	6285752522528312052810	14212319911346526380505	22069087302806417183220	22634143547525814001392	14049953747105373355968	4286362930788981888000	282257335822049280000				86215965170904167779375
12	26483088952989760000	801913424446228980000	11019339086427191826000	90704433102024441301200	496973383532442859282920	1903442681535230691126954	5202413350166798402022585	10155959907524683905994020	13905499737690655028990640	12780554061866332104434880	7201524569988385660594176	2014571268870162610176000	121935169075125288960000			53885426298951667403449375
13	4896926239069662200000	162784333237129140240000	2476626741492927547050000	22803237186233581080990000	141520240018405565801730900	623734338955094675482525488	2002266019382585194089682263	4719110075416434380198858775	8112268922336469022015444860	9939541010103674021113315920	8280298331943155430578229312	4273184137356149358977759232	1103937742781032263985152000	61821130721088521502720000		39283129494201291141175898750
14	1042533363892498293400000	37749460010064588534480000	630070473122366035193730000	6418427151983528211854310000	44522383389933160722160167300	222085427602980234583039432176	819662185391896878813085866939	2266836022986715976699771163099	4699828427746665951129104865360	7221785860950081527849954626080	8010745681165276580605448930304	6104608932728196885699807480576	2906662677616525491940475068416	697644980371301460602904576000	36350824864000050643599360000	33037820694432054050623227456250

Technical details

Node id: n-2894b5b080f5601a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-bdd17ebb32a9b330

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:              5,               3;
  2:             49,             111,               36;
  3:            760,            3486,             4329,               972;
  4:          17720,          127422,           308745,            270972,             46656;
  5:         588220,         5843472,         21761595,          36527247,          25035156,           3499200;
  6:       26537740,       338490384,       1722444399,        4389787683,        5685472296,        3211180848,          377913600;
  7:     1567290800,     24486210960,     158929027632,      549405579276,     1071717884433,     1135047259368,       546492563856,       55553299200;
  8:   117573742000,   2177743539600,   17239805555760,    75661171641612,   199160142593553,   315899350655400,    283714028584848,   119203770154752,    10666233446400;
  9: 10932603994000, 234424251709200, 2193812144678520, 11706260938350624, 38985019550388801, 83125199704261707, 111351093204715848, 86948576426610864, 32433042017684736, 2591894727475200;

%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) = 1, P(1,x) = 5 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
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] = 1; v[2] = 5 + 3*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 + (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.