A088617 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(3 - n\right)\,P_{n-2}(x) + \left(-9 - 12\,x + 6\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

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

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	2														4
3	6	14	8													28
4	22	82	100	40												244
5	90	462	876	728	224											2380
6	394	2578	6664	8512	5376	1344										24868
7	1806	14366	47120	81600	78720	40128	8448									272188
8	8558	80162	318924	698760	910800	706464	302016	54912								3080596
9	41586	448286	2097620	5565560	9159920	9577568	6214208	2288000	366080							35758828
10	206098	2512930	13522960	42159040	83923840	110638528	96608512	53886976	17425408	2489344						423373636
11	1037718	14119470	85910040	307844160	719496960	1146180672	1260499968	945045504	462339072	133293056	17199104					5092965724
12	5293446	79506546	539721780	2185932840	5869325280	10970948352	14568433152	13744573440	9029529600	3934295040	1023346688	120393728				62071299892
13	27297738	448596462	3361254204	15185025240	46068792000	98885504256	153995702784	175324689408	144839608320	84680780800	33261082624	7881158656	852017152			764811509644
14	142078746	2535709746	20786912280	103639789440	350649953280	850186541952	1519855383552	2028409405440	2021151052800	1485124439040	782243033088	279705059328	60858368000	6085836800		9511373563492
15	745387038	14356922430	127817433120	697172611200	2602861198080	7036578586752	14205365611008	21756109799424	25402891717632	22504214926336	14890399473664	7136252231680	2341830000640	471043768320	43818024960	119231457692284

Technical details

Node id: n-36f587956712a55a · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8419344160e8e665

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,       2;
   3:      6,      14,        8;
   4:     22,      82,      100,       40;
   5:     90,     462,      876,      728,      224;
   6:    394,    2578,     6664,     8512,     5376,      1344;
   7:   1806,   14366,    47120,    81600,    78720,     40128,     8448;
   8:   8558,   80162,   318924,   698760,   910800,    706464,   302016,    54912;
   9:  41586,  448286,  2097620,  5565560,  9159920,   9577568,  6214208,  2288000,   366080;
  10: 206098, 2512930, 13522960, 42159040, 83923840, 110638528, 96608512, 53886976, 17425408, 2489344;

%F Formula

The row polynomials satisfy n*P(n,x) = -(n-3)*P(n-2,x) + (6*n-9+(8*n-12)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 3)*P[n - 2, x] + (6*n - 9 + (8*n - 12)*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 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(n - 3)*v[i - 2] + (6*n - 9 + (8*n - 12)*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

Related ?

Same fingerprint ?

A253284 → basis A013609

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.