A088617 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(3 - n\right)\,P_{n-2}(x) + \left(-9 - 24\,x + 6\,n + 16\,n x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	4														6
3	6	28	32													66
4	22	164	400	320												906
5	90	924	3504	5824	3584											13926
6	394	5156	26656	68096	86016	43008										229326
7	1806	28732	188480	652800	1259520	1284096	540672									3956106
8	8558	160324	1275696	5590080	14572800	22606848	19329024	7028736								70572066
9	41586	896572	8390480	44524480	146558720	306482176	397709312	292864000	93716480							1291183806
10	206098	5025860	54091840	337272320	1342781440	3540432896	6182944768	6897532928	4460904448	1274544128						24095736726
11	1037718	28238940	343640160	2462753280	11511951360	36677781504	80671997952	120965824512	118358802432	68246044672	17611882496					456879955026
12	5293446	159013092	2158887120	17487462720	93909204480	351070347264	932379721728	1759305400320	2311559577600	2014359060480	1047907008512	246566354944				8776867331706
13	27297738	897192924	13445016816	121480201920	737100672000	3164336136192	9855724978176	22441560244224	37078939729920	43356559769600	34059348606976	16140612927488	3489862254592			170459895028566
14	142078746	5071419492	83147649120	829118315520	5610399252480	27205969342464	97270744547328	259636403896320	517414669516800	760383712788480	801016865882112	572835961503744	249275875328000	49855175065600		3341423256586206
15	745387038	28713844860	511269732480	5577380889600	41645779169280	225170514776064	909143399104512	2784782054326272	6503140279713792	11522158042284032	15247769061031936	14615044570480640	9592135682621440	3858790550077440	717914520944640	66023812564384026

Technical details

Node id: n-39310b120a1d8578 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-2eec2e552ea6c0fc

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      1;
   2:      2,       4;
   3:      6,      28,       32;
   4:     22,     164,      400,       320;
   5:     90,     924,     3504,      5824,       3584;
   6:    394,    5156,    26656,     68096,      86016,      43008;
   7:   1806,   28732,   188480,    652800,    1259520,    1284096,     540672;
   8:   8558,  160324,  1275696,   5590080,   14572800,   22606848,   19329024,    7028736;
   9:  41586,  896572,  8390480,  44524480,  146558720,  306482176,  397709312,  292864000,   93716480;
  10: 206098, 5025860, 54091840, 337272320, 1342781440, 3540432896, 6182944768, 6897532928, 4460904448, 1274544128;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n-3)*P(n-2,x) + (6*n-9+(16*n-24)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 3)*P[n - 2, x] + (6*n - 9 + (16*n - 24)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(n - 3)*v[i - 2] + (6*n - 9 + (16*n - 24)*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[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.