A142462 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 + 7\,n\right)\,P_{n}(x) = \left(6 - 156\,x - 78\,x^{2} + 42\,n - 1008\,n x - 413\,n x^{2} + 588\,n^{2} x + 882\,n^{2} x^{2} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(42\,x - 42\,x^{3} + 294\,n x - 49\,n x^{2} - 245\,n x^{3} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-12 + 72\,x - 35\,n - 168\,n x + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-42\,x + 42\,x^{2} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	32	3													36
3	1	286	429	4												720
4	1	2332	12870	4664	5											19872
5	1	18714	270006	360008	46785	6										695520
6	1	149784	4864149	15840352	8106915	449352	7									29410560
7	1	1198358	81050571	536568172	670710215	162101142	4194253	8								1455822720
8	1	9586964	1293052656	15694927192	38901192470	23542390788	3017122864	38347856	9							82496620800
9	1	76695826	20114812140	419423417168	1834406305270	2201287566324	733990980044	53639499040	345131217	10						5263284407040
10	1	613566736	308279675223	10568302653696	75741791525610	158492704775328	106038508135854	21136605307392	924839025669	3067833680	11					373214712499200
11	1	4908534030	4683097535001	255931913698260	2857030870098690	9614013814633284	11216349450405498	4571249392157904	575846805821085	15610325116670	26996937165	12				29110747574937600
12	1	39268272396	70769221899210	6030126424348472	101089044083863035	517526349532719624	964606052734872828	690035132710292832	181960279350953463	15075316060871180	259487146963770	235609634376	13			2476652832144691200
13	1	314146179338	1066132716358482	139362267675458264	3413992392379905775	25545439198514138058	71667097323674843004	81905254084199820576	38318158797771207087	6827984784759811550	383246236107510226	4264530865433928	2041950165697	14		228205868104760832000
14	1	2513169434888	16032044383242825	3176847691361363936	111374180635214188405	1181830426881285733608	4780456891619208430029	8190525408419982057600	6146301717796125124323	1969717378135476222680	245023197397471214491	9530543074084091808	69472192327385575	17592186044216	15	22638022115992274534400

Technical details

Node id: n-5fa42daebc5aeec3 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-0c87c216373988a4

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: 1,        2;
  2: 1,       32,           3;
  3: 1,      286,         429,            4;
  4: 1,     2332,       12870,         4664,             5;
  5: 1,    18714,      270006,       360008,         46785,             6;
  6: 1,   149784,     4864149,     15840352,       8106915,        449352,            7;
  7: 1,  1198358,    81050571,    536568172,     670710215,     162101142,      4194253,           8;
  8: 1,  9586964,  1293052656,  15694927192,   38901192470,   23542390788,   3017122864,    38347856,         9;
  9: 1, 76695826, 20114812140, 419423417168, 1834406305270, 2201287566324, 733990980044, 53639499040, 345131217, 10;

%F Formula

The row polynomials satisfy (7*n-6)*P(n,x) = (42*n+6+(588*n^2-1008*n-156)*x-(343*n^3-882*n^2+413*n+78)*x^2)*P(n-2,x) + ((294*n+42)*x-(343*n^2+49*n)*x^2+(343*n^2-245*n-42)*x^3)*d/dx P(n-2,x) - (35*n+12-(98*n^2-168*n+72)*x)*P(n-1,x) + ((49*n-42)*x-(49*n-42)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((42*n + 6 + (588*n^2 - 1008*n - 156)*x - (343*n^3 - 882*n^2 + 413*n + 78)*x^2)*P[n - 2, x] + ((294*n + 42)*x - (343*n^2 + 49*n)*x^2 + (343*n^2 - 245*n - 42)*x^3)*D[P[n - 2, x], x] - (35*n + 12 - (98*n^2 - 168*n + 72)*x)*P[n - 1, x] + ((49*n - 42)*x - (49*n - 42)*x^2)*D[P[n - 1, x], x])/(7*n - 6)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((42*n + 6 + (588*n^2 - 1008*n - 156)*x - (343*n^3 - 882*n^2 + 413*n + 78)*x^2)*v[i - 2] + ((294*n + 42)*x - (343*n^2 + 49*n)*x^2 + (343*n^2 - 245*n - 42)*x^3)*deriv(v[i - 2], x) - (35*n + 12 - (98*n^2 - 168*n + 72)*x)*v[i - 1] + ((49*n - 42)*x - (49*n - 42)*x^2)*deriv(v[i - 1], x))/(7*n - 6)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*…?
    
    ↑ Liu–Wang theorem (above)

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.