A060187 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 12\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(3 - 20\,x + 585\,x^{2} - 8\,n - 152\,n x - 1824\,n x^{2} + 4\,n^{2} + 144\,n^{2} x + 1976\,n^{2} x^{2} - 32\,n^{3} x - 896\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(-52\,x^{2} + 612\,x^{3} + 32\,n x + 8\,n x^{2} - 1752\,n x^{3} - 16\,n^{2} x - 48\,n^{2} x^{2} + 1664\,n^{2} x^{3} + 32\,n^{3} x^{2} - 608\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} - 104\,x^{3} + 108\,x^{4} + 32\,n x^{2} + 256\,n x^{3} - 288\,n x^{4} - 16\,n^{2} x^{2} - 224\,n^{2} x^{3} + 240\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 108\,x - 4\,n + 216\,n x - 144\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(20\,x + 36\,x^{2} - 40\,n x - 120\,n x^{2} + 16\,n^{2} x + 112\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	4							5
3	1	24	9						34
4	1	92	207	16					316
5	1	304	2070	1216	25				3616
6	1	948	15138	26912	5925	36			48960
7	1	2888	94887	376768	263575	25992	49		764160
8	1	8716	545913	4155568	6493075	2183652	106771	64	13493760

All 15 rows

Technical details

Node id: n-f47e83b3f1520fc5 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5d3f47477444bdf9

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,     4;
   3: 1,    24,        9;
   4: 1,    92,      207,        16;
   5: 1,   304,     2070,      1216,         25;
   6: 1,   948,    15138,     26912,       5925,         36;
   7: 1,  2888,    94887,    376768,     263575,      25992,         49;
   8: 1,  8716,   545913,   4155568,    6493075,    2183652,     106771,        64;
   9: 1, 26208,  2984508,  39764608,  116875350,   89470368,   16248988,    419328,      81;
  10: 1, 78692, 15807060, 347327552, 1735332350, 2498878584, 1063690628, 112405760, 1593513, 100;

%F Formula

The row polynomials satisfy (4*n^2-12*n+9)*P(n,x) = (4*n^2-8*n+3-(32*n^3-144*n^2+152*n+20)*x+(144*n^4-896*n^3+1976*n^2-1824*n+585)*x^2)*P(n-2,x) - ((16*n^2-32*n)*x-(32*n^3-48*n^2+8*n-52)*x^2-(64*n^4-608*n^3+1664*n^2-1752*n+612)*x^3)*d/dx P(n-2,x) - ((16*n^2-32*n+4)*x^2-(64*n^3-224*n^2+256*n-104)*x^3+(64*n^3-240*n^2+288*n-108)*x^4)*d^2/dx^2 P(n-2,x) - (4*n-6-(32*n^3-144*n^2+216*n-108)*x)*P(n-1,x) + ((16*n^2-40*n+20)*x-(32*n^3-112*n^2+120*n-36)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 152*n + 20)*x + (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*P[n - 2, x] - ((16*n^2 - 32*n)*x - (32*n^3 - 48*n^2 + 8*n - 52)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*D[P[n - 2, x], x] - ((16*n^2 - 32*n + 4)*x^2 - (64*n^3 - 224*n^2 + 256*n - 104)*x^3 + (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 6 - (32*n^3 - 144*n^2 + 216*n - 108)*x)*P[n - 1, x] + ((16*n^2 - 40*n + 20)*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 152*n + 20)*x + (144*n^4 - 896*n^3 + 1976*n^2 - 1824*n + 585)*x^2)*v[i - 2] - ((16*n^2 - 32*n)*x - (32*n^3 - 48*n^2 + 8*n - 52)*x^2 - (64*n^4 - 608*n^3 + 1664*n^2 - 1752*n + 612)*x^3)*deriv(v[i - 2], x) - ((16*n^2 - 32*n + 4)*x^2 - (64*n^3 - 224*n^2 + 256*n - 104)*x^3 + (64*n^3 - 240*n^2 + 288*n - 108)*x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 6 - (32*n^3 - 144*n^2 + 216*n - 108)*x)*v[i - 1] + ((16*n^2 - 40*n + 20)*x - (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 12*n + 9)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
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: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
  - 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: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.