A144439 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 12\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(204\,x^{2} - 288\,n x^{2} + 144\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(96\,x^{3} - 128\,n x^{3} + 64\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(72\,x - 420\,x^{2} + 64\,n x + 1128\,n x^{2} - 32\,n^{2} x - 1040\,n^{2} x^{2} + 288\,n^{3} x^{2}\right)\,P_{n-3}(x) + \left(72\,x^{2} - 504\,x^{3} - 64\,n x^{2} + 928\,n x^{3} + 32\,n^{2} x^{2} - 672\,n^{2} x^{3} + 128\,n^{3} x^{3}\right)\,P_{n-3}'(x) + \left(96\,x^{3} - 96\,x^{4} - 128\,n x^{3} + 128\,n x^{4} + 64\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-3}''(x) + \left(3 - 36\,x - 135\,x^{2} - 8\,n - 120\,n x - 216\,n x^{2} + 4\,n^{2} + 144\,n^{2} x + 792\,n^{2} x^{2} - 32\,n^{3} x - 608\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(24\,x - 84\,x^{2} - 180\,x^{3} + 32\,n x - 24\,n x^{2} - 408\,n x^{3} - 16\,n^{2} x - 48\,n^{2} x^{2} + 928\,n^{2} x^{3} + 32\,n^{3} x^{2} - 480\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(12\,x^{2} + 24\,x^{3} - 36\,x^{4} + 32\,n x^{2} + 64\,n x^{3} - 96\,n x^{4} - 16\,n^{2} x^{2} - 160\,n^{2} x^{3} + 176\,n^{2} x^{4} + 64\,n^{3} x^{3} - 64\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(6 - 36\,x - 4\,n + 120\,n x - 112\,n^{2} x + 32\,n^{3} x\right)\,P_{n-1}(x) + \left(12\,x - 36\,x^{2} - 40\,n x - 24\,n x^{2} + 16\,n^{2} x + 80\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 1 + 32\,x + 9\,x^{2}\)

\(P_{3}(x) = 1 + 124\,x + 279\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	32	9						42
3	1	124	279	16					420
4	1	408	2934	1632	25				5000
5	1	1268	21654	38496	7925	36			69380
6	1	3856	135783	549376	377175	34704	49		1100944
7	1	11628	780057	6095152	9523675	3120228	142443	64	19673248

All 15 rows

Technical details

Node id: n-4ad248c935e4a6fc · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-dbdf092354a1f6be

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,      4;
  2: 1,     32,        9;
  3: 1,    124,      279,        16;
  4: 1,    408,     2934,      1632,         25;
  5: 1,   1268,    21654,     38496,       7925,         36;
  6: 1,   3856,   135783,    549376,     377175,      34704,         49;
  7: 1,  11628,   780057,   6095152,    9523675,    3120228,     142443,        64;
  8: 1,  34952,  4257756,  58403296,  173142950,  131407416,   23181116,    559232,      81;
  9: 1, 104932, 22520736, 509998080, 2581140550, 3716842392, 1561869120, 160147456, 2124873, 100;

%F Formula

The row polynomials satisfy (4*n^2-12*n+9)*P(n,x) = (144*n^2-288*n+204)*x^2*P(n-4,x) + (64*n^2-128*n+96)*x^3*d/dx P(n-4,x) - ((32*n^2-64*n-72)*x-(288*n^3-1040*n^2+1128*n-420)*x^2)*P(n-3,x) + ((32*n^2-64*n+72)*x^2+(128*n^3-672*n^2+928*n-504)*x^3)*d/dx P(n-3,x) + ((64*n^2-128*n+96)*x^3-(64*n^2-128*n+96)*x^4)*d^2/dx^2 P(n-3,x) + (4*n^2-8*n+3-(32*n^3-144*n^2+120*n+36)*x+(144*n^4-608*n^3+792*n^2-216*n-135)*x^2)*P(n-2,x) - ((16*n^2-32*n-24)*x-(32*n^3-48*n^2-24*n-84)*x^2-(64*n^4-480*n^3+928*n^2-408*n-180)*x^3)*d/dx P(n-2,x) - ((16*n^2-32*n-12)*x^2-(64*n^3-160*n^2+64*n+24)*x^3+(64*n^3-176*n^2+96*n+36)*x^4)*d^2/dx^2 P(n-2,x) - (4*n-6-(32*n^3-112*n^2+120*n-36)*x)*P(n-1,x) + ((16*n^2-40*n+12)*x-(32*n^3-80*n^2+24*n+36)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 32*x + 9*x^2, P(3,x) = 1 + 124*x + 279*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 32*x + 9*x^2;
P[3, x_] := 1 + 124*x + 279*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((144*n^2 - 288*n + 204)*x^2*P[n - 4, x] + (64*n^2 - 128*n + 96)*x^3*D[P[n - 4, x], x] - ((32*n^2 - 64*n - 72)*x - (288*n^3 - 1040*n^2 + 1128*n - 420)*x^2)*P[n - 3, x] + ((32*n^2 - 64*n + 72)*x^2 + (128*n^3 - 672*n^2 + 928*n - 504)*x^3)*D[P[n - 3, x], x] + ((64*n^2 - 128*n + 96)*x^3 - (64*n^2 - 128*n + 96)*x^4)*D[P[n - 3, x], {x, 2}] + (4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 120*n + 36)*x + (144*n^4 - 608*n^3 + 792*n^2 - 216*n - 135)*x^2)*P[n - 2, x] - ((16*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 24*n - 84)*x^2 - (64*n^4 - 480*n^3 + 928*n^2 - 408*n - 180)*x^3)*D[P[n - 2, x], x] - ((16*n^2 - 32*n - 12)*x^2 - (64*n^3 - 160*n^2 + 64*n + 24)*x^3 + (64*n^3 - 176*n^2 + 96*n + 36)*x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 6 - (32*n^3 - 112*n^2 + 120*n - 36)*x)*P[n - 1, x] + ((16*n^2 - 40*n + 12)*x - (32*n^3 - 80*n^2 + 24*n + 36)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 12*n + 9)]];
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 + 4*x; v[3] = 1 + 32*x + 9*x^2; v[4] = 1 + 124*x + 279*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((144*n^2 - 288*n + 204)*x^2*v[i - 4] + (64*n^2 - 128*n + 96)*x^3*deriv(v[i - 4], x) - ((32*n^2 - 64*n - 72)*x - (288*n^3 - 1040*n^2 + 1128*n - 420)*x^2)*v[i - 3] + ((32*n^2 - 64*n + 72)*x^2 + (128*n^3 - 672*n^2 + 928*n - 504)*x^3)*deriv(v[i - 3], x) + ((64*n^2 - 128*n + 96)*x^3 - (64*n^2 - 128*n + 96)*x^4)*deriv(deriv(v[i - 3], x), x) + (4*n^2 - 8*n + 3 - (32*n^3 - 144*n^2 + 120*n + 36)*x + (144*n^4 - 608*n^3 + 792*n^2 - 216*n - 135)*x^2)*v[i - 2] - ((16*n^2 - 32*n - 24)*x - (32*n^3 - 48*n^2 - 24*n - 84)*x^2 - (64*n^4 - 480*n^3 + 928*n^2 - 408*n - 180)*x^3)*deriv(v[i - 2], x) - ((16*n^2 - 32*n - 12)*x^2 - (64*n^3 - 160*n^2 + 64*n + 24)*x^3 + (64*n^3 - 176*n^2 + 96*n + 36)*x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 6 - (32*n^3 - 112*n^2 + 120*n - 36)*x)*v[i - 1] + ((16*n^2 - 40*n + 12)*x - (32*n^3 - 80*n^2 + 24*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 - 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 T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m …?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.