A316649 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 8\,x - 2\,x^{2} + 12\,n + 12\,n x + 3\,n x^{2} - 4\,n^{2} - 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 14\,x + 4\,x^{2} - 30\,n - 35\,n x - 10\,n x^{2} + 12\,n^{2} + 14\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 5\,x + 2\,x^{2}\)

\(P_{2}(x) = 26 + 75\,x + 79\,x^{2} + 36\,x^{3} + 6\,x^{4}\)

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	x²⁸	P_n(1)
0	1																													1
1	3	5	2																											10
2	26	75	79	36	6																									222
3	252	1032	1731	1523	742	190	20																							5490
4	2568	13620	31218	40407	32321	16370	5130	910	70																					142614
5	26928	175344	508968	867536	961947	725305	376756	133182	30674	4158	252																			3811050
6	287648	2220400	7796032	16467480	23312570	23308331	16880935	8925714	3420480	926730	168546	18480	924																	103734270
7	3112896	27787520	114434384	288220656	496093796	617421272	573096675	403113863	215964092	87711732	26588802	5834730	876348	80652	3432															2860340850
8	34013312	344688192	1628462176	4761771728	9647028552	14360465644	16250282594	14261536527	9811595521	5309937444	2253379464	742072254	185932362	34270236	4382664	347490	12870													79630179030
9	374416128	4246329600	22633498368	75344428032	175555769376	304080653856	405817542480	426726453216	358446658899	242349243437	132215131006	58071350260	20384348446	5641629994	1204132072	191268220	21294130	1482910	48620											2233305679050
10	4145895936	52024566528	308786893056	1152772998912	3036086660544	5997007439520	9218823584832	11294794398384	11202571870938	9084409559211	6056536358159	3325839190520	1501779170430	554636952870	165914819070	39585736476	7357080588	1026577266	101183082	6281704	184756									63004207402782
11	46127840256	634519296000	4149995468544	17169502053120	50428296063360	111855021929472	194590547226720	272167537023648	311234938450380	294317812145000	231861160304067	152806984768795	84366971296818	38973571588770	15003086405970	4778163718926	1244906727480	260965159104	42943555330	5342261730	472353908	26453700	705432							1785938892796530
12	515268544512	7709895398400	55088129407488	250153961325312	810548999834112	1994550761960064	3873246845709120	6088746912368736	7885772413675464	8518959995691684	7742448761256594	5953573086368775	3885928571075201	2155290834740134	1014818981719114	404377063894232	135611076237780	37952865175368	8756396781132	1636626753762	241607915406	27113866780	2173553564	110870396	2704156					50825958456827286
13	5775088103424	93383480193024	723692637714432	3578312423264256	12678400417092864	34272365695611648	73480487178371328	128239317280402944	185516962200910032	225381459607117968	232114040321488248	203993585480812848	153670794617219019	99481211507068097	55391683742449328	26507480019763554	10875342483447244	3808777038265012	1131249095055816	282312108261744	58436637235482	9853211768862	1318798738936	134795728228	9882858132	462826700	10400600			1451296386222169770
14	64912164888576	1127993351270400	9423381717344256	50395315119212544	193814901309998592	570889471661812992	1339306217806652160	2569238313455354112	4105849852587651168	5540516117879927184	6376384147138317312	6304583833672629960	5383664902938575162	3984560728824152043	2561304419183044999	1430957745477341446	694399864149772560	292113093132398876	106156991580317964	33155769721369152	8835690961834308	1989310636408810	373344070141458	57335145345452	7018456105856	658587536100	44485594900	1925596800	40116600	41559171377180711742

Technical details

Node id: n-451dfe842bdfe4bd · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-fbcf47e112a39ff9

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:         3,          5,           2;
  2:        26,         75,          79,          36,            6;
  3:       252,       1032,        1731,        1523,          742,          190,           20;
  4:      2568,      13620,       31218,       40407,        32321,        16370,         5130,          910,           70;
  5:     26928,     175344,      508968,      867536,       961947,       725305,       376756,       133182,        30674,         4158,          252;
  6:    287648,    2220400,     7796032,    16467480,     23312570,     23308331,     16880935,      8925714,      3420480,       926730,       168546,       18480,         924;
  7:   3112896,   27787520,   114434384,   288220656,    496093796,    617421272,    573096675,    403113863,    215964092,     87711732,     26588802,     5834730,      876348,      80652,       3432;
  8:  34013312,  344688192,  1628462176,  4761771728,   9647028552,  14360465644,  16250282594,  14261536527,   9811595521,   5309937444,   2253379464,   742072254,   185932362,   34270236,    4382664,    347490,    12870;
  9: 374416128, 4246329600, 22633498368, 75344428032, 175555769376, 304080653856, 405817542480, 426726453216, 358446658899, 242349243437, 132215131006, 58071350260, 20384348446, 5641629994, 1204132072, 191268220, 21294130, 1482910, 48620;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(4*n^2-12*n+8)*x+(n^2-3*n+2)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(14*n^2-35*n+14)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 5*x + 2*x^2, P(2,x) = 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 5*x + 2*x^2;
P[2, x_] := 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (4*n^2 - 12*n + 8)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (14*n^2 - 35*n + 14)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];
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] = 3 + 5*x + 2*x^2; v[3] = 26 + 75*x + 79*x^2 + 36*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (4*n^2 - 12*n + 8)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (14*n^2 - 35*n + 14)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1])/(n^2 - 2*n)); 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 Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] order…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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