A200536 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 - 4\,x^{2} + 7\,n x + 8\,n x^{2}\right)\,P_{n}(x) = \left(18 + 84\,x + 146\,x^{2} + 112\,x^{3} + 32\,x^{4} + 42\,n x + 146\,n x^{2} + 168\,n x^{3} + 64\,n x^{4}\right)\,P_{n-1}(x)\)

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

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	28	24													58
2	36	336	876	896	320											2464
3	216	3024	13176	27104	29280	16128	3584									92512
4	1296	24192	147744	456960	825680	913920	612864	229376	36864							3248896
5	7776	181440	1425600	5886720	14920800	24838464	27852160	20930560	10137600	2867200	360448					109408768
6	46656	1306368	12550464	64834560	211550400	470340864	739920832	836161536	676961280	384204800	145440768	33030144	3407872			3579756544
7	279936	9144576	103856256	644184576	2571246720	7154106624	14491123584	21872671744	24841926144	21197352960	13408575488	6107824128	1896480768	359661568	31457280	114689892352

All 15 rows

Technical details

Node id: n-9df7bf46a00d9e2f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7c392c1c35db3029

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,        28,         24;
  2:       36,       336,        876,         896,          320;
  3:      216,      3024,      13176,       27104,        29280,         16128,          3584;
  4:     1296,     24192,     147744,      456960,       825680,        913920,        612864,        229376,          36864;
  5:     7776,    181440,    1425600,     5886720,     14920800,      24838464,      27852160,      20930560,       10137600,        2867200,         360448;
  6:    46656,   1306368,   12550464,    64834560,    211550400,     470340864,     739920832,     836161536,      676961280,      384204800,      145440768,       33030144,        3407872;
  7:   279936,   9144576,  103856256,   644184576,   2571246720,    7154106624,   14491123584,   21872671744,    24841926144,    21197352960,    13408575488,     6107824128,     1896480768,     359661568,      31457280;
  8:  1679616,  62705664,  821892096,  5950070784,  28085460480,   93851799552,  232595112960,  439404478464,   642597693696,   732340797440,   649789521920,   444927049728,   230786203648,   87757422592,   23089643520,    3758096384,    285212672;
  9: 10077696, 423263232, 6288482304, 52045701120, 283597562880, 1106347382784, 3243040561152, 7359034540032, 13168613574144, 18784053693440, 21459962861568, 19624092106752, 14276242046976, 8158742839296, 3585233387520, 1169707499520, 266959060992, 38050725888, 2550136832;

%F Formula

The row polynomials satisfy (3+7*n*x+(8*n-4)*x^2)*P(n,x) = (18+(42*n+84)*x+(146*n+146)*x^2+(168*n+112)*x^3+(64*n+32)*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[((18 + (42*n + 84)*x + (146*n + 146)*x^2 + (168*n + 112)*x^3 + (64*n + 32)*x^4)*P[n - 1, x])/(3 + 7*n*x + (8*n - 4)*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] = ((18 + (42*n + 84)*x + (146*n + 146)*x^2 + (168*n + 112)*x^3 + (64*n + 32)*x^4)*v[i - 1])/(3 + 7*n*x + (8*n - 4)*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_A013609 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.