A123519 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(27 - 75\,x + 12\,x^{2} + 18\,n - 149\,n x + 44\,n x^{2} - 99\,n^{2} x + 63\,n^{2} x^{2} - 22\,n^{3} x + 44\,n^{3} x^{2} + 15\,n^{4} x^{2} + 2\,n^{5} x^{2}\right)\,P_{n}(x) = \left(27 + 69\,x - 264\,x^{2} + 36\,x^{3} + 18\,n - 77\,n x - 398\,n x^{2} + 114\,n x^{3} - 99\,n^{2} x - 165\,n^{2} x^{2} + 136\,n^{2} x^{3} - 22\,n^{3} x + 6\,n^{3} x^{2} + 76\,n^{3} x^{3} + 15\,n^{4} x^{2} + 20\,n^{4} x^{3} + 2\,n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(36\,x - 138\,x^{2} - 42\,x^{3} + 18\,x^{4} - 176\,n x^{2} + 8\,n x^{3} + 48\,n x^{4} - 44\,n^{2} x^{2} + 66\,n^{2} x^{3} + 44\,n^{2} x^{4} + 32\,n^{3} x^{3} + 16\,n^{3} x^{4} + 4\,n^{4} x^{3} + 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 24\,x + 36\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	24	36															61
1	1	48	180	128														357
2	1	80	540	896	400													1917
3	1	120	1260	3584	3600	1152												9717
4	1	168	2520	10752	18000	12672	3136											47249
5	1	224	4536	26880	66000	76032	40768	8192										222633
6	1	288	7560	59136	198000	329472	285376	122880	20736									1023449
7	1	360	11880	118272	514800	1153152	1426880	983040	352512	51200								4612097
8	1	440	17820	219648	1201200	3459456	5707520	5570560	3172608	972800	123904							20445957
9	1	528	25740	384384	2574000	9225216	19405568	25067520	20093184	9728000	2601984	294912						89401037
10	1	624	36036	640640	5148000	22404096	58216704	95256576	100465920	68096000	28621824	6782976	692224					386361621
11	1	728	49140	1025024	9724000	50409216	158016768	317521920	421956864	374528000	219433984	81395712	17305600	1605632				1652972589
12	1	840	65520	1584128	17503200	106419456	395041920	952565760	1547175168	1722828800	1316603904	678297600	224972800	43352064	3686400			7010097561
13	1	960	85680	2376192	30232800	212838912	921764480	2619555840	5083575552	6891315200	6583019520	4408934400	2024755200	606928896	106905600	8388608		29500677841
14	1	1088	110160	3472896	50388000	406328832	2027881856	6694420480	15250726656	24611840000	28526417920	23808245760	14173286400	5866979328	1603584000	260046848	18939904	123302670129

Computed in your browser ?

Technical details

Node id: n-35b4d678c1c4e937 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-32b6b38366ac538d

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1,  24,    36;
  1: 1,  48,   180,    128;
  2: 1,  80,   540,    896,     400;
  3: 1, 120,  1260,   3584,    3600,    1152;
  4: 1, 168,  2520,  10752,   18000,   12672,     3136;
  5: 1, 224,  4536,  26880,   66000,   76032,    40768,     8192;
  6: 1, 288,  7560,  59136,  198000,  329472,   285376,   122880,    20736;
  7: 1, 360, 11880, 118272,  514800, 1153152,  1426880,   983040,   352512,   51200;
  8: 1, 440, 17820, 219648, 1201200, 3459456,  5707520,  5570560,  3172608,  972800,  123904;
  9: 1, 528, 25740, 384384, 2574000, 9225216, 19405568, 25067520, 20093184, 9728000, 2601984, 294912;

%F Formula

The row polynomials satisfy (18*n+27-(22*n^3+99*n^2+149*n+75)*x+(2*n^5+15*n^4+44*n^3+63*n^2+44*n+12)*x^2)*P(n,x) = (18*n+27-(22*n^3+99*n^2+77*n-69)*x+(2*n^5+15*n^4+6*n^3-165*n^2-398*n-264)*x^2+(2*n^5+20*n^4+76*n^3+136*n^2+114*n+36)*x^3)*P(n-1,x) + (36*x-(44*n^2+176*n+138)*x^2+(4*n^4+32*n^3+66*n^2+8*n-42)*x^3+(2*n^4+16*n^3+44*n^2+48*n+18)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 24*x + 36*x^2.

%t Mathematica

P[0, x_] := 1 + 24*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 27 - (22*n^3 + 99*n^2 + 77*n - 69)*x + (2*n^5 + 15*n^4 + 6*n^3 - 165*n^2 - 398*n - 264)*x^2 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^3)*P[n - 1, x] + (36*x - (44*n^2 + 176*n + 138)*x^2 + (4*n^4 + 32*n^3 + 66*n^2 + 8*n - 42)*x^3 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^4)*D[P[n - 1, x], x])/(18*n + 27 - (22*n^3 + 99*n^2 + 149*n + 75)*x + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 24*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18*n + 27 - (22*n^3 + 99*n^2 + 77*n - 69)*x + (2*n^5 + 15*n^4 + 6*n^3 - 165*n^2 - 398*n - 264)*x^2 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^3)*v[i - 1] + (36*x - (44*n^2 + 176*n + 138)*x^2 + (4*n^4 + 32*n^3 + 66*n^2 + 8*n - 42)*x^3 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^4)*deriv(v[i - 1], x))/(18*n + 27 - (22*n^3 + 99*n^2 + 149*n + 75)*x + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*x^2)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-ac14b3745c4c6962
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-ac14b3745c4c6962
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-ac14b3745c4c6962
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-ac14b3745c4c6962

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image euler_polynomial, shift 0 e-ac14b3745c4c6962

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-2a4594def0262eef root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

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.
[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