A121574 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 - 104\,x - 66\,x^{2} - 4\,x^{3} + 5\,x^{4} + 507\,n x - 78\,n x^{2} - 117\,n x^{3} - 6\,n x^{4} + 442\,n^{2} x^{2} + 62\,n^{2} x^{3} - 31\,n^{2} x^{4} - 2\,n^{2} x^{5} + 117\,n^{3} x^{3} + 12\,n^{3} x^{4} - 3\,n^{3} x^{5} + 9\,n^{4} x^{4}\right)\,P_{n}(x) = \left(169 + 572\,x + 701\,x^{2} + 356\,x^{3} + 47\,x^{4} - 16\,x^{5} - 5\,x^{6} + 507\,n x + 1313\,n x^{2} + 1164\,n x^{3} + 362\,n x^{4} - 9\,n x^{5} - 13\,n x^{6} + 442\,n^{2} x^{2} + 855\,n^{2} x^{3} + 472\,n^{2} x^{4} + 48\,n^{2} x^{5} - 11\,n^{2} x^{6} + 117\,n^{3} x^{3} + 165\,n^{3} x^{4} + 45\,n^{3} x^{5} - 3\,n^{3} x^{6} + 9\,n^{4} x^{4} + 9\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(507 + 364\,x - 250\,x^{2} - 384\,x^{3} - 123\,x^{4} - 16\,x^{5} - 2\,x^{6} + 1521\,n x + 1430\,n x^{2} + 303\,n x^{3} - 127\,n x^{4} - 48\,n x^{5} - 7\,n x^{6} + 1326\,n^{2} x^{2} + 1096\,n^{2} x^{3} + 215\,n^{2} x^{4} - 25\,n^{2} x^{5} - 8\,n^{2} x^{6} + 351\,n^{3} x^{3} + 207\,n^{3} x^{4} + 21\,n^{3} x^{5} - 3\,n^{3} x^{6} + 27\,n^{4} x^{4} + 9\,n^{4} x^{5}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	4							7
2	10	28	9						47
3	33	140	99	16					288
4	109	620	684	240	25				1678
5	360	2568	3861	2128	475	36			9428
6	1189	10200	19440	14704	5150	828	49		51560
7	3927	39364	90909	87664	42225	10620	1323	64	276096

All 15 rows

Technical details

Node id: n-dd50126698bba20c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-585b39f92b176d60

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:     3,      4;
  2:    10,     28,       9;
  3:    33,    140,      99,      16;
  4:   109,    620,     684,     240,      25;
  5:   360,   2568,    3861,    2128,     475,     36;
  6:  1189,  10200,   19440,   14704,    5150,    828,     49;
  7:  3927,  39364,   90909,   87664,   42225,  10620,   1323,    64;
  8: 12970, 148756,  403686,  473872,  291775, 100908,  19600,  1984,   81;
  9: 42837, 553220, 1725237, 2388560, 1794950, 794304, 211925, 33344, 2835, 100;

%F Formula

The row polynomials satisfy (169+(507*n-104)*x+(442*n^2-78*n-66)*x^2+(117*n^3+62*n^2-117*n-4)*x^3+(9*n^4+12*n^3-31*n^2-6*n+5)*x^4-(3*n^3+2*n^2)*x^5)*P(n,x) = (169+(507*n+572)*x+(442*n^2+1313*n+701)*x^2+(117*n^3+855*n^2+1164*n+356)*x^3+(9*n^4+165*n^3+472*n^2+362*n+47)*x^4+(9*n^4+45*n^3+48*n^2-9*n-16)*x^5-(3*n^3+11*n^2+13*n+5)*x^6)*P(n-2,x) + (507+(1521*n+364)*x+(1326*n^2+1430*n-250)*x^2+(351*n^3+1096*n^2+303*n-384)*x^3+(27*n^4+207*n^3+215*n^2-127*n-123)*x^4+(9*n^4+21*n^3-25*n^2-48*n-16)*x^5-(3*n^3+8*n^2+7*n+2)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((169 + (507*n + 572)*x + (442*n^2 + 1313*n + 701)*x^2 + (117*n^3 + 855*n^2 + 1164*n + 356)*x^3 + (9*n^4 + 165*n^3 + 472*n^2 + 362*n + 47)*x^4 + (9*n^4 + 45*n^3 + 48*n^2 - 9*n - 16)*x^5 - (3*n^3 + 11*n^2 + 13*n + 5)*x^6)*P[n - 2, x] + (507 + (1521*n + 364)*x + (1326*n^2 + 1430*n - 250)*x^2 + (351*n^3 + 1096*n^2 + 303*n - 384)*x^3 + (27*n^4 + 207*n^3 + 215*n^2 - 127*n - 123)*x^4 + (9*n^4 + 21*n^3 - 25*n^2 - 48*n - 16)*x^5 - (3*n^3 + 8*n^2 + 7*n + 2)*x^6)*P[n - 1, x])/(169 + (507*n - 104)*x + (442*n^2 - 78*n - 66)*x^2 + (117*n^3 + 62*n^2 - 117*n - 4)*x^3 + (9*n^4 + 12*n^3 - 31*n^2 - 6*n + 5)*x^4 - (3*n^3 + 2*n^2)*x^5)]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((169 + (507*n + 572)*x + (442*n^2 + 1313*n + 701)*x^2 + (117*n^3 + 855*n^2 + 1164*n + 356)*x^3 + (9*n^4 + 165*n^3 + 472*n^2 + 362*n + 47)*x^4 + (9*n^4 + 45*n^3 + 48*n^2 - 9*n - 16)*x^5 - (3*n^3 + 11*n^2 + 13*n + 5)*x^6)*v[i - 2] + (507 + (1521*n + 364)*x + (1326*n^2 + 1430*n - 250)*x^2 + (351*n^3 + 1096*n^2 + 303*n - 384)*x^3 + (27*n^4 + 207*n^3 + 215*n^2 - 127*n - 123)*x^4 + (9*n^4 + 21*n^3 - 25*n^2 - 48*n - 16)*x^5 - (3*n^3 + 8*n^2 + 7*n + 2)*x^6)*v[i - 1])/(169 + (507*n - 104)*x + (442*n^2 - 78*n - 66)*x^2 + (117*n^3 + 62*n^2 - 117*n - 4)*x^3 + (9*n^4 + 12*n^3 - 31*n^2 - 6*n + 5)*x^4 - (3*n^3 + 2*n^2)*x^5)); 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_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    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 reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (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.