A272866 → D → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 10\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	P_n(1)
0																													0
1	5	4																											9
2	50	140	120	32																									342
3	375	1800	3300	2880	1200	192																							9747
4	2500	17000	48000	72800	64000	32640	8960	1024																					246924
5	15625	137500	525000	1140000	1550000	1368000	784000	281600	57600	5120																			5864445
6	93750	1012500	4875000	13800000	25500000	32280000	28560000	17664000	7488000	2073600	337920	24576																	133709346
7	546875	7000000	40687500	142100000	332500000	550200000	662200000	586880000	383040000	181888000	61107200	13762560	1863680	114688															2963890503
8	3125000	46250000	315000000	1309000000	3710000000	7593600000	11592000000	13436800000	11923200000	8099840000	4178944000	1608499200	447283200	84869120	9830400	524288													64358765208
9	17578125	295312500	2306250000	11115000000	37012500000	90342000000	167328000000	240163200000	270396000000	240163200000	168284160000	92510208000	39416832000	12747571200	3022848000	495452160	50135040	2359296											1375668606321
10	97656250	1835937500	16171875000	88687500000	339375000000	962550000000	2097900000000	3594960000000	4914360000000	5406032000000	4805152000000	3451161600000	1994803200000	919941120000	333619200000	92995584000	19218432000	2772172800	249036800	10485760									29041892800110
11	537109375	11171875000	109570312500	673750000000	2913281250000	9417375000000	23619750000000	47084400000000	75804300000000	99608960000000	107517872000000	95624601600000	70076864000000	42187622400000	20731392000000	8229027840000	2596577280000	635830272000	116424704000	14994636800	1211105280	46137344							606975559522299
12	2929687500	66796875000	721875000000	4919062500000	23718750000000	86088750000000	244282500000000	555588000000000	1030194000000000	1575736800000000	2003886720000000	2129168256000000	1894583808000000	1411860172800000	879098880000000	455137689600000	194398617600000	67702947840000	18901893120000	4126408704000	678218956800	78894858240	5788139520	201326592					12580947961007652
13	15869140625	393554687500	4646484375000	34754687500000	184894531250000	744583125000000	2358606250000000	6028880000000000	12657645000000000	22103224000000000	32389042400000000	40069960320000000	42011827520000000	37398629632000000	28266800640000000	18106961100800000	9793079296000000	4444972646400000	1678227865600000	520501329920000	130284650496000	25655823564800	3824995532800	405673082880	27262976000	872415232			258957845530740837
14	85449218750	2290039062500	29326171875000	238875000000000	1389882812500000	6150178125000000	21515243750000000	61071010000000000	143243100000000000	281301020000000000	466962496000000000	659780284800000000	797262211200000000	826421361920000000	735934348800000000	563012509696000000	369494687744000000	207397616025600000	99091120128000000	40023497113600000	13536220938240000	3783371784192000	858115801088000	153894046924800	20992491520000	2045813719040	126835752960	3758096384	5298675916244389434

Technical details

Node id: n-dff119f470326354 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-dc7db2765c5515ed

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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:        5,         4;
  2:       50,       140,        120,          32;
  3:      375,      1800,       3300,        2880,        1200,         192;
  4:     2500,     17000,      48000,       72800,       64000,       32640,         8960,         1024;
  5:    15625,    137500,     525000,     1140000,     1550000,     1368000,       784000,       281600,        57600,         5120;
  6:    93750,   1012500,    4875000,    13800000,    25500000,    32280000,     28560000,     17664000,      7488000,      2073600,       337920,       24576;
  7:   546875,   7000000,   40687500,   142100000,   332500000,   550200000,    662200000,    586880000,    383040000,    181888000,     61107200,    13762560,     1863680,      114688;
  8:  3125000,  46250000,  315000000,  1309000000,  3710000000,  7593600000,  11592000000,  13436800000,  11923200000,   8099840000,   4178944000,  1608499200,   447283200,    84869120,    9830400,    524288;
  9: 17578125, 295312500, 2306250000, 11115000000, 37012500000, 90342000000, 167328000000, 240163200000, 270396000000, 240163200000, 168284160000, 92510208000, 39416832000, 12747571200, 3022848000, 495452160, 50135040, 2359296;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n+10*n*x+4*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5 + 4*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 10*n*x + 4*n*x^2)*P[n - 1, x])/(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] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 10*n*x + 4*n*x^2)*v[i - 1])/(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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272866 → basis A013609 → D

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.