A316649 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 24\,x - 18\,x^{2} + 12\,n + 36\,n x + 27\,n x^{2} - 4\,n^{2} - 12\,n^{2} x - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 42\,x + 36\,x^{2} - 30\,n - 105\,n x - 90\,n x^{2} + 12\,n^{2} + 42\,n^{2} x + 36\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 15\,x + 18\,x^{2}\)

\(P_{2}(x) = 26 + 225\,x + 711\,x^{2} + 972\,x^{3} + 486\,x^{4}\)

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	15	18																											36
2	26	225	711	972	486																									2420
3	252	3096	15579	41121	60102	46170	14580																							180900
4	2568	40860	280962	1090989	2618001	3977910	3739770	1990170	459270																					14200500
5	26928	526032	4580712	23423472	77917707	176249115	274655124	291269034	201252114	81841914	14880348																			1146622500
6	287648	6661200	70164288	444621960	1888318170	5663924433	12306201615	19520536518	22441769280	18240826590	9952472754	3273676560	491051484																	94300512500
7	3112896	83362560	1029909456	7781957712	40183597476	150033369096	417787476075	881610018381	1416940407612	1726430020956	1570042169298	1033604915310	465727257468	128585338596	16415149608															7856258062500
8	34013312	1034064576	14656159584	128567836656	781409312712	3489593151492	11846456011026	31189980384549	64373878213281	104515498710252	133059803969736	131455873579338	98812080393642	54637825470228	20962146049416	4986101693430	554011299270													660808950312500
9	374416128	12738988800	203701485312	2034299556864	14220017319456	73891598887008	295840988467920	933250753183392	2351768529036339	4770160158570471	7807171270773294	10287165484508220	10833078522490686	8994580456924062	5759326372281768	2744489900835540	916642473047730	191503239114330	18836384175180											55994177264062500
10	4145895936	156073699584	2779082037504	31124870970624	245923019504064	1457272807803360	6720522393342528	24701715349265808	73500074045224218	178808433353950113	357632415412930791	589162435083046440	798107024112489630	884270450610557010	793565436252418830	568012051220631732	316698195446151948	132572355463334358	39200399106967098	7300982506299768	644204338791156									4772633954695312500
11	46127840256	1903557888000	37349959216896	463576555434240	4084691981132160	27180770328861696	141856508928278880	595230403470718176	2042012431172943180	5793057496450035000	13691169654794852283	27069298930837727865	44836067592952254738	62136461576122552710	71759297184075924930	68561426833643313882	53589152568854593080	33701083184011493952	16637213205347156370	6209104955407757910	1646996238165789108	276715045526201100	22137203642096088							408740049351914062500
12	515268544512	23129686195200	495793164667392	6754156955783424	65654468986563072	484675835156295552	2823596950521948480	13316089497350425632	51738552806124719304	167678689595199416172	457183856903440619106	1054657611530969384925	2065141765740775894641	3436229749515394659282	4853847730174088969466	5802368882751392804424	5837612163317445319380	4901239195064047104984	3392407522824185413548	1902188211758869888854	842434710595768381806	283620623217888296340	68208413955238396476	10437691517248305492	763733525652315036					35144456395140820312500
13	5775088103424	280150440579072	6513233739429888	96614435428134912	1026950433784521984	8328184864033630464	53567275153032698112	280459386892241238528	1217176789000170719952	4436183269446902964144	13706101966943559556152	36136851687169553584656	81666960762169492676379	158605183573583329613331	264936706197939119894832	380353365607945398335478	468147833664400630686924	491866087552200886876956	438269077587331717014024	328120485100157401418448	203755955159574401316282	103068075286252917364986	41385301838824645236024	12690098347684295828556	2791211041328241513492	392147791056092528100	26436929734118597400			3031914725363680664062500
14	64912164888576	3383980053811200	84810435456098304	1360673508218738688	15699007006109885952	138726141613820557056	976354232781049424640	5618924191526859442944	26938480882827579313248	109053978748230606762672	376519107504370498956288	1116838112383605379524120	2861100259682579322668442	6352676814861108557651889	12250639636515509555822031	20532679610784043015899522	29891637214493161603775760	37723532461552171427656788	41127393588815668388364396	38535673555872691852065984	30808149417780551481029508	20808891887320864100918430	11715932519775770560169922	5397712841328924327144804	1982219304789154349732736	558013717658660563392300	113076413529196837922100	14683826115179015241600	917739132198688452600	262311031642032231445312500

Technical details

Node id: n-207567e2aa72e118 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-25b337cc6c0a5b46

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,          15,           18;
  2:        26,         225,          711,           972,            486;
  3:       252,        3096,        15579,         41121,          60102,          46170,           14580;
  4:      2568,       40860,       280962,       1090989,        2618001,        3977910,         3739770,         1990170,           459270;
  5:     26928,      526032,      4580712,      23423472,       77917707,      176249115,       274655124,       291269034,        201252114,         81841914,         14880348;
  6:    287648,     6661200,     70164288,     444621960,     1888318170,     5663924433,     12306201615,     19520536518,      22441769280,      18240826590,       9952472754,        3273676560,         491051484;
  7:   3112896,    83362560,   1029909456,    7781957712,    40183597476,   150033369096,    417787476075,    881610018381,    1416940407612,    1726430020956,    1570042169298,     1033604915310,      465727257468,     128585338596,      16415149608;
  8:  34013312,  1034064576,  14656159584,  128567836656,   781409312712,  3489593151492,  11846456011026,  31189980384549,   64373878213281,  104515498710252,  133059803969736,   131455873579338,    98812080393642,   54637825470228,   20962146049416,    4986101693430,    554011299270;
  9: 374416128, 12738988800, 203701485312, 2034299556864, 14220017319456, 73891598887008, 295840988467920, 933250753183392, 2351768529036339, 4770160158570471, 7807171270773294, 10287165484508220, 10833078522490686, 8994580456924062, 5759326372281768, 2744489900835540, 916642473047730, 191503239114330, 18836384175180;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(12*n^2-36*n+24)*x+(9*n^2-27*n+18)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(42*n^2-105*n+42)*x+(36*n^2-90*n+36)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 15*x + 18*x^2, P(2,x) = 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 15*x + 18*x^2;
P[2, x_] := 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (12*n^2 - 36*n + 24)*x + (9*n^2 - 27*n + 18)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (42*n^2 - 105*n + 42)*x + (36*n^2 - 90*n + 36)*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 + 15*x + 18*x^2; v[3] = 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (12*n^2 - 36*n + 24)*x + (9*n^2 - 27*n + 18)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (42*n^2 - 105*n + 42)*x + (36*n^2 - 90*n + 36)*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