A367270 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n - 3\,n^{2} + 3\,n^{2} x + 2\,n^{3} - 6\,n^{3} x + 3\,n^{4} x\right)\,P_{n}(x) = \left(-2 - 12\,x - 18\,x^{2} + 10\,n + 48\,n x + 45\,n x^{2} - 16\,n^{2} - 54\,n^{2} x - 18\,n^{2} x^{2} + 8\,n^{3} + 6\,n^{3} x - 36\,n^{3} x^{2} + 12\,n^{4} x + 27\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-x - 6\,x^{2} - 9\,x^{3} + 4\,n x + 18\,n x^{2} + 18\,n x^{3} - 4\,n^{2} x - 12\,n^{2} x^{2} - 6\,n^{3} x^{2} - 18\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	3	12													15
3	10	72	81												163
4	35	360	810	432											1637
5	126	1680	5670	6480	2025										15981
6	462	7560	34020	60480	42525	8748									153795
7	1716	33264	187110	453600	510300	244944	35721								1466655
8	6435	144144	972972	2993760	4677750	3674160	1285956	139968							13895145
9	24310	617760	4864860	18162144	36486450	40415760	23575860	6298560	531441						130977145
10	92378	2625480	23629320	103783680	255405150	367783416	306486180	138568320	29229255	1968300					1229571479
11	352716	11085360	112239270	567103680	1654052400	2942267328	3218104890	2101619520	759960630	129907800	7144929				11503838523
12	1352078	46558512	523783260	2993047200	10101534300	21436519104	29177484336	25219434240	13299311025	3940536600	557304462	25509168			107322374285
13	5200300	194699232	2409402996	15364308960	58925616750	145462093920	237588086736	257238229248	180870629940	78810732000	19505656170	2321334288	89813529		998785804069
14	20058300	811246800	10951831800	77100895872	331292911950	933381769320	1781910650520	2327393502720	2061925181316	1205804199600	442128206520	92853371520	9430420545	312487308	9275316734091

Technical details

Node id: n-948b1cd4a1f67fa8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-d70ec49f0fd8d1b9

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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;
  2:     3,     12;
  3:    10,     72,      81;
  4:    35,    360,     810,      432;
  5:   126,   1680,    5670,     6480,     2025;
  6:   462,   7560,   34020,    60480,    42525,     8748;
  7:  1716,  33264,  187110,   453600,   510300,   244944,    35721;
  8:  6435, 144144,  972972,  2993760,  4677750,  3674160,  1285956,  139968;
  9: 24310, 617760, 4864860, 18162144, 36486450, 40415760, 23575860, 6298560, 531441;

%F Formula

The row polynomials satisfy (2*n^3-3*n^2+n+(3*n^4-6*n^3+3*n^2)*x)*P(n,x) = (8*n^3-16*n^2+10*n-2+(12*n^4+6*n^3-54*n^2+48*n-12)*x+(27*n^4-36*n^3-18*n^2+45*n-18)*x^2)*P(n-1,x) - ((4*n^2-4*n+1)*x+(6*n^3+12*n^2-18*n+6)*x^2+(18*n^3-18*n+9)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 16*n^2 + 10*n - 2 + (12*n^4 + 6*n^3 - 54*n^2 + 48*n - 12)*x + (27*n^4 - 36*n^3 - 18*n^2 + 45*n - 18)*x^2)*P[n - 1, x] - ((4*n^2 - 4*n + 1)*x + (6*n^3 + 12*n^2 - 18*n + 6)*x^2 + (18*n^3 - 18*n + 9)*x^3)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + n + (3*n^4 - 6*n^3 + 3*n^2)*x)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^3 - 16*n^2 + 10*n - 2 + (12*n^4 + 6*n^3 - 54*n^2 + 48*n - 12)*x + (27*n^4 - 36*n^3 - 18*n^2 + 45*n - 18)*x^2)*v[i - 1] - ((4*n^2 - 4*n + 1)*x + (6*n^3 + 12*n^2 - 18*n + 6)*x^2 + (18*n^3 - 18*n + 9)*x^3)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + n + (3*n^4 - 6*n^3 + 3*n^2)*x)); 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 ≥ 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 basis_map_A013610 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023