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 | Pn(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 |
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 ExampleTriangle 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 FormulaThe 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 MathematicaP[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/GPP(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-ac14b3745c4c6962Laguerre’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-8d057b494ddd47e8root: theorem checkThree-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
-
theorem_check three_term_recurrence
e-2a4594def0262eefroot: theorem checkThree-term recurrence criterion [Sze39, §3.3; Fav35] 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…
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