A008955 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x + 16\,n^{3} x^{2} - 8\,n^{4} - 28\,n^{4} x - 24\,n^{4} x^{2} + 4\,n^{5} + 16\,n^{5} x + 16\,n^{5} x^{2} - n^{6} - 4\,n^{6} x - 4\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 4\,n^{2} x - 4\,n^{3} - 8\,n^{3} x + 2\,n^{4} + 4\,n^{4} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	2														6
2	36	58	16													110
3	540	1696	1492	288												4016
4	12380	60192	99732	61856	9216											243376
5	406880	2717992	6840432	7844456	3793216	460800										22063776
6	18236960	155974024	533582640	918662600	812373376	323338752	33177600									2795345952
7	1072117200	11212876160	48793837088	113378178048	149366349968	107958297344	36597482496	3251404800								471630543104
8	80154178000	992832393600	5261089412640	15478982484992	27383373794448	29327529831168	17968717480960	5311998885888	416179814400							102220858276096
9	7433680696000	106521171847200	666605634113280	2380886058728384	5315102351827776	7615771204173600	6887203437643264	3667690920609792	962091694227456	67421129932800						27676727283799552
10	837968570296000	13644943814967200	98583735940001280	414913550186734784	1121766119358573376	2023064742044541600	2440812264832900864	1912933214900590592	901769750205382656	212712098955264000	13484225986560000					9154522614795812352
11	112884822648312000	2059607276377062400	16884262521043533760	81910815141708284928	260508291012871641152	567926030529976609280	860683586627561520960	897962098214061698560	621573517543414018048	263023656681943498752	56384586206281728000	3263182688747520000				3632292519266635427840
12	17914959444702840000	362118052759404160000	3322486324565195224000	18266924549155988659200	66879798460994433135680	171277783500231835705856	313305246108788862139200	409929012003145027171840	377086925513479977594880	233892435701214829117440	89827576193767572504576	17651718931636813824000	939796614359285760000			1702759736913543927836672
13	3308743850307084800000	73407380809010225040000	745520500151300118240000	4583277035913053278320000	18998616489174084036441600	55952703400180692902514048	120097413077864666704896768	189435611335087923698033280	218248970887764663563750400	179646783040556773847982080	100988926731872758744547328	35516978902923677886578688	6443697678987760238592000	317651255653438586880000		931052866460790110916616192
14	703715135041012265600000	17003416601485252256080000	189408510949109205276000000	1287975864439893999706800000	5965326632888262447173875200	19874440849054475431158708096	49014176628778203230434699008	90633666989100258470622804096	125756568659967789553667665920	129505216717691217384488222720	96502391026197235873981136896	49621015556476820005606391808	16099242864059949473329053696	2713026079743077341200384000	124519292216147926056960000	587304682803298966607224381440

Technical details

Node id: n-68bb4b80c7a32a0f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-02ab16b23749367b

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 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:             4,               2;
  2:            36,              58,              16;
  3:           540,            1696,            1492,              288;
  4:         12380,           60192,           99732,            61856,             9216;
  5:        406880,         2717992,         6840432,          7844456,          3793216,           460800;
  6:      18236960,       155974024,       533582640,        918662600,        812373376,        323338752,         33177600;
  7:    1072117200,     11212876160,     48793837088,     113378178048,     149366349968,     107958297344,      36597482496,       3251404800;
  8:   80154178000,    992832393600,   5261089412640,   15478982484992,   27383373794448,   29327529831168,   17968717480960,    5311998885888,    416179814400;
  9: 7433680696000, 106521171847200, 666605634113280, 2380886058728384, 5315102351827776, 7615771204173600, 6887203437643264, 3667690920609792, 962091694227456, 67421129932800;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(4*n^6-16*n^5+28*n^4-24*n^3+8*n^2)*x+(4*n^6-16*n^5+24*n^4-16*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(4*n^4-8*n^3+4*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];
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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*v[i - 1])/(n^2 - 2*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 Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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
[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.