A272866 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 9\,x^{2} + 15\,n x + 18\,n x^{2}\right)\,P_{n}(x) = \left(25 + 150\,x + 315\,x^{2} + 270\,x^{3} + 81\,x^{4} + 75\,n x + 315\,n x^{2} + 405\,n x^{3} + 162\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	30	27													62
2	25	300	945	1080	405											2755
3	125	2250	12150	29700	36450	21870	5103									107648
4	625	15000	114750	432000	921375	1166400	867510	349920	59049							3926629
5	3125	93750	928125	4725000	14428125	28248750	36358875	30618000	16238475	4920750	649539					137212514
6	15625	562500	6834375	43875000	174656250	464737500	857941875	1115370000	1018595250	639697500	263063295	63772920	6908733			4656030823
7	78125	3281250	47250000	366187500	1798453125	6059812500	14623284375	25861275000	33842458125	32722987500	23074872975	11532269700	3868890480	781218270	71744535	154654063460

All 15 rows

Technical details

Node id: n-cdeab012a595d9e0 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f80803402873efeb

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:       5,        30,         27;
  2:      25,       300,        945,        1080,          405;
  3:     125,      2250,      12150,       29700,        36450,        21870,          5103;
  4:     625,     15000,     114750,      432000,       921375,      1166400,        867510,        349920,          59049;
  5:    3125,     93750,     928125,     4725000,     14428125,     28248750,      36358875,      30618000,       16238475,        4920750,         649539;
  6:   15625,    562500,    6834375,    43875000,    174656250,    464737500,     857941875,    1115370000,     1018595250,      639697500,      263063295,       63772920,        6908733;
  7:   78125,   3281250,   47250000,   366187500,   1798453125,   6059812500,   14623284375,   25861275000,    33842458125,    32722987500,    23074872975,    11532269700,     3868890480,      781218270,      71744535;
  8:  390625,  18750000,  312187500,  2835000000,  16567031250,  67614750000,  201823650000,  452709000000,   774833596875,  1018595250000,  1027570698000,   788658444000,   452176574850,   187492384800,   53090955900,    9183300480,    731794257;
  9: 1953125, 105468750, 1993359375, 20756250000, 140674218750, 674552812500, 2401120968750, 6534756000000, 13849020309375, 23099845781250, 30467844680625, 31758914160000, 26006198195250, 16522766410500, 7974405065250, 2823864897600, 691545572865, 104603532030, 7360989291;

%F Formula

The row polynomials satisfy (5+15*n*x+(18*n-9)*x^2)*P(n,x) = (25+(75*n+150)*x+(315*n+315)*x^2+(405*n+270)*x^3+(162*n+81)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((25 + (75*n + 150)*x + (315*n + 315)*x^2 + (405*n + 270)*x^3 + (162*n + 81)*x^4)*P[n - 1, x])/(5 + 15*n*x + (18*n - 9)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((25 + (75*n + 150)*x + (315*n + 315)*x^2 + (405*n + 270)*x^3 + (162*n + 81)*x^4)*v[i - 1])/(5 + 15*n*x + (18*n - 9)*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.