A128813 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 24\,n x - n^{2} - 12\,n^{2} x + 81\,n^{2} x^{2} - 3\,n^{3} - 12\,n^{3} x - 81\,n^{3} x^{2} + n^{4} + 12\,n^{4} x - 81\,n^{4} x^{2} - n^{5} - 12\,n^{5} x + 81\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-24\,n x^{2} + 12\,n^{2} x^{2} + 36\,n^{2} x^{3} + 12\,n^{3} x^{2} - 36\,n^{3} x^{3} - 12\,n^{4} x^{2} - 36\,n^{4} x^{3} + 12\,n^{5} x^{2} + 36\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(4\,n - 24\,n x + 4\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(24\,n x^{2} - 24\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2 + 12\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	12														14
2	8	120	243													371
3	56	1416	6561	7776												15809
4	616	22296	167751	435456	364500											990619
5	9856	467616	4941486	20387376	36450000	23619600										85875934
6	216832	12771264	174997260	1001968704	2808782325	3826375200	2025380700									9850492285
7	6288128	443222208	7488689436	55190016576	212963079825	450715190832	496218271500	222213196800								1445237955305
8	232660736	19115692992	384784505676	3479858985024	17192949250725	49796480235168	84615209097804	78219045273600	30373766337600							264081442039325
9	10702393856	1004958675072	23506479007416	252421794673344	1524908420186850	5632947425158668	13042402861711104	18517982339831040	14761650440073600	5062294389600000						58819129811310950
10	599334055936	63341265748992	1689453732535776	21030854343882624	150472365469685100	677763296445280968	1995440669489372679	3847777913472768000	4693727955454150500	3290491353240000000	1010687074883640000					15689144877131120575
11	40155381747712	4718537377483776	141411933971072928	2003754954892728960	16588069049107588500	88312821704550440568	316351733247791190369	773846511300008733312	1278708869519744539500	1367818643111427900000	856051952426443080000	238154627099854080000				4937983153040550585625
12	3172275158069248	410349890137076736	13655852712959969376	217124005968491925888	2043080360262153517500	12566228661446777825832	53119420639474819360059	157821293925739891298304	330197481554253030045432	477462457333729004400000	454912274852258698926000	257206997267842406400000	65403214467297426720000			1810963642443150955534375
13	291849314542370816	41216314365222678528	1508395869609016567680	26603049065791162318848	280580351940052960746600	1959268788079361350776144	9556387832280948574907502	33460426743472234159830720	84907891312052000893545312	155215326894840200753445600	199572102817458502324644000	171460658203120790415072000	88425145959786120925440000	20707663903645862183040000		765573605759775154485383750
14	30936027341491306496	4736659459036991152128	189102024984907329580800	3664624887954912483698688	42835205517714701542947600	334953739176420326176929024	1853013603110806728085466952	7478576228660789948904000000	22339967547197731697654565897	49472560161095506071799521600	80314964744618943763588054440	92989627982753263829004672000	72759927551203546024668570000	34498968063474006396944640000	7488039179443369807260000000	369577292500760525493925105625

Technical details

Node id: n-bbcc67c907ed26d9 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-cde818948dd99780

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:           2,            12;
  2:           8,           120,            243;
  3:          56,          1416,           6561,            7776;
  4:         616,         22296,         167751,          435456,           364500;
  5:        9856,        467616,        4941486,        20387376,         36450000,         23619600;
  6:      216832,      12771264,      174997260,      1001968704,       2808782325,       3826375200,        2025380700;
  7:     6288128,     443222208,     7488689436,     55190016576,     212963079825,     450715190832,      496218271500,      222213196800;
  8:   232660736,   19115692992,   384784505676,   3479858985024,   17192949250725,   49796480235168,    84615209097804,    78219045273600,    30373766337600;
  9: 10702393856, 1004958675072, 23506479007416, 252421794673344, 1524908420186850, 5632947425158668, 13042402861711104, 18517982339831040, 14761650440073600, 5062294389600000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(12*n^5-12*n^4+12*n^3+12*n^2-24*n)*x-(81*n^5-81*n^4-81*n^3+81*n^2)*x^2)*P(n-2,x) + ((12*n^5-12*n^4+12*n^3+12*n^2-24*n)*x^2+(36*n^5-36*n^4-36*n^3+36*n^2)*x^3)*d/dx P(n-2,x) + (4*n^3+4*n+(24*n^3-24*n)*x)*P(n-1,x) - (24*n^3-24*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x - (81*n^5 - 81*n^4 - 81*n^3 + 81*n^2)*x^2)*P[n - 2, x] + ((12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x^2 + (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^3)*D[P[n - 2, x], x] + (4*n^3 + 4*n + (24*n^3 - 24*n)*x)*P[n - 1, x] - (24*n^3 - 24*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];
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] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x - (81*n^5 - 81*n^4 - 81*n^3 + 81*n^2)*x^2)*v[i - 2] + ((12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x^2 + (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^3)*deriv(v[i - 2], x) + (4*n^3 + 4*n + (24*n^3 - 24*n)*x)*v[i - 1] - (24*n^3 - 24*n)*x^2*deriv(v[i - 1], x))/(4*n - 4)); 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 basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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
[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.