A091042 → (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

\(\left(-27 + 60\,x - 3\,x^{2} + 18\,n - 175\,n x + 20\,n x^{2} + 156\,n^{2} x - 51\,n^{2} x^{2} - 44\,n^{3} x + 62\,n^{3} x^{2} - 36\,n^{4} x^{2} + 8\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-27 - 264\,x - 177\,x^{2} - 12\,x^{3} + 18\,n + 41\,n x - 137\,n x^{2} - 10\,n x^{3} + 156\,n^{2} x + 473\,n^{2} x^{2} + 63\,n^{2} x^{3} - 44\,n^{3} x - 166\,n^{3} x^{2} - 22\,n^{3} x^{3} - 36\,n^{4} x^{2} - 44\,n^{4} x^{3} + 8\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(36\,x + 18\,x^{2} - 48\,x^{3} - 6\,x^{4} + 48\,n x^{2} - 40\,n x^{3} - 8\,n x^{4} - 88\,n^{2} x^{2} + 60\,n^{2} x^{3} + 28\,n^{2} x^{4} + 16\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	12														13
3	1	40	45													86
4	1	84	315	112												512
5	1	144	1134	1344	225											2848
6	1	220	2970	7392	4125	396										15104
7	1	312	6435	27456	32175	10296	637									77312
8	1	420	12285	80080	160875	108108	22295	960								385024
9	1	544	21420	198016	607750	700128	303212	43520	1377							1875968
10	1	684	34884	434112	1889550	3325608	2469012	744192	78489	1900						8978432
11	1	840	53865	868224	5087250	12697776	14402570	7441920	1648269	133000	2541					42336256
12	1	1012	79695	1615152	12257850	41186376	66251822	52300160	19857717	3364900	214291	3312				197132288
13	1	1200	113850	2833600	27039375	117675360	254814700	285273600	165480975	48070000	6428730	331200	4225			908066816
14	1	1404	157950	4736160	55501875	303706260	851809140	1283731200	1056069495	468682500	107451630	11625120	494325	5292		4143972352
15	1	1624	213759	7600320	107303625	721080360	2542900815	4963760640	5496977115	3459729000	1211815605	224752320	20069595	716184	6525	18756927488

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Technical details

Node id: n-f93cc9bcf835d9b8 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d8c33c6b14bc1f96

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,  12;
   3: 1,  40,    45;
   4: 1,  84,   315,    112;
   5: 1, 144,  1134,   1344,     225;
   6: 1, 220,  2970,   7392,    4125,     396;
   7: 1, 312,  6435,  27456,   32175,   10296,     637;
   8: 1, 420, 12285,  80080,  160875,  108108,   22295,    960;
   9: 1, 544, 21420, 198016,  607750,  700128,  303212,  43520,  1377;
  10: 1, 684, 34884, 434112, 1889550, 3325608, 2469012, 744192, 78489, 1900;

%F Formula

The row polynomials satisfy (18*n-27-(44*n^3-156*n^2+175*n-60)*x+(8*n^5-36*n^4+62*n^3-51*n^2+20*n-3)*x^2)*P(n,x) = (18*n-27-(44*n^3-156*n^2-41*n+264)*x+(8*n^5-36*n^4-166*n^3+473*n^2-137*n-177)*x^2+(24*n^5-44*n^4-22*n^3+63*n^2-10*n-12)*x^3)*P(n-1,x) + (36*x-(88*n^2-48*n-18)*x^2+(16*n^4+60*n^2-40*n-48)*x^3-(16*n^4-28*n^2+8*n+6)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((18*n - 27 - (44*n^3 - 156*n^2 - 41*n + 264)*x + (8*n^5 - 36*n^4 - 166*n^3 + 473*n^2 - 137*n - 177)*x^2 + (24*n^5 - 44*n^4 - 22*n^3 + 63*n^2 - 10*n - 12)*x^3)*P[n - 1, x] + (36*x - (88*n^2 - 48*n - 18)*x^2 + (16*n^4 + 60*n^2 - 40*n - 48)*x^3 - (16*n^4 - 28*n^2 + 8*n + 6)*x^4)*D[P[n - 1, x], x])/(18*n - 27 - (44*n^3 - 156*n^2 + 175*n - 60)*x + (8*n^5 - 36*n^4 + 62*n^3 - 51*n^2 + 20*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((18*n - 27 - (44*n^3 - 156*n^2 - 41*n + 264)*x + (8*n^5 - 36*n^4 - 166*n^3 + 473*n^2 - 137*n - 177)*x^2 + (24*n^5 - 44*n^4 - 22*n^3 + 63*n^2 - 10*n - 12)*x^3)*v[i - 1] + (36*x - (88*n^2 - 48*n - 18)*x^2 + (16*n^4 + 60*n^2 - 40*n - 48)*x^3 - (16*n^4 - 28*n^2 + 8*n + 6)*x^4)*deriv(v[i - 1], x))/(18*n - 27 - (44*n^3 - 156*n^2 + 175*n - 60)*x + (8*n^5 - 36*n^4 + 62*n^3 - 51*n^2 + 20*n - 3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-14374e20b2874ad7
Coefficients of one sign. Every nonzero row, n >= 1: 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-14374e20b2874ad7
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-14374e20b2874ad7
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-14374e20b2874ad7

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-14374e20b2874ad7

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-1c0df90858517049 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