A200536 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 6\,x^{2} + 7\,n x + 12\,n x^{2}\right)\,P_{n}(x) = \left(12 + 84\,x + 219\,x^{2} + 252\,x^{3} + 108\,x^{4} + 42\,n x + 219\,n x^{2} + 378\,n x^{3} + 216\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	42	54													102
2	36	504	1971	3024	1620											7155
3	216	4536	29646	91476	148230	122472	40824									437400
4	1296	36288	332424	1542240	4180005	6940080	6980904	3919104	944784							24877125
5	7776	272160	3207600	19867680	75536550	188617086	317253510	357618240	259815600	110224800	20785248					1353206250
6	46656	1959552	28238544	218816640	1070973900	3571650936	8428160727	14286603744	17349777180	14770123200	8386847568	2857026816	442158912			71412384375
7	279936	13716864	233676576	2174122944	13016936520	54326497176	165062954574	373715102376	636671396214	814897457640	773206029288	528311875392	246061434528	69997156992	9183300480	3686871937500

All 15 rows

Technical details

Node id: n-c4e761abf0fd75f8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-a42848b2e2dea42b

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:        6,        42,          54;
  2:       36,       504,        1971,         3024,          1620;
  3:      216,      4536,       29646,        91476,        148230,        122472,          40824;
  4:     1296,     36288,      332424,      1542240,       4180005,       6940080,        6980904,         3919104,          944784;
  5:     7776,    272160,     3207600,     19867680,      75536550,     188617086,      317253510,       357618240,       259815600,       110224800,         20785248;
  6:    46656,   1959552,    28238544,    218816640,    1070973900,    3571650936,     8428160727,     14286603744,     17349777180,     14770123200,       8386847568,       2857026816,        442158912;
  7:   279936,  13716864,   233676576,   2174122944,   13016936520,   54326497176,   165062954574,    373715102376,    636671396214,    814897457640,     773206029288,     528311875392,     246061434528,      69997156992,       9183300480;
  8:  1679616,  94058496,  1849257216,  20081488896,  142182643680,  712687102848,  2649403708560,   7507637456256,  16469076048201,  28153640460960,   37470138163920,   38485103553792,   29943664759008,   17079306306048,    6740542552320,   1645647446016,    187339329792;
  9: 10077696, 634894848, 14149085184, 175654241280, 1435712662080, 8401325438016, 36940258891872, 125736004211328, 337497162734214, 722122126656210, 1237489596692118, 1697436056852928, 1852290124435296, 1587850507785024, 1046634530656320, 512207767572480, 175349612685312, 37489905879552, 3768826516992;

%F Formula

The row polynomials satisfy (2+7*n*x+(12*n-6)*x^2)*P(n,x) = (12+(42*n+84)*x+(219*n+219)*x^2+(378*n+252)*x^3+(216*n+108)*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[((12 + (42*n + 84)*x + (219*n + 219)*x^2 + (378*n + 252)*x^3 + (216*n + 108)*x^4)*P[n - 1, x])/(2 + 7*n*x + (12*n - 6)*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] = ((12 + (42*n + 84)*x + (219*n + 219)*x^2 + (378*n + 252)*x^3 + (216*n + 108)*x^4)*v[i - 1])/(2 + 7*n*x + (12*n - 6)*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 of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    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

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.