A121574 → (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

\(\left(-32\,n + 96\,n x + 16\,n x^{3} - 32\,n^{2} + 32\,n^{2} x + 120\,n^{2} x^{2} + 16\,n^{2} x^{3} - 4\,n^{2} x^{4} - 72\,n^{3} x + 42\,n^{3} x^{2} + 39\,n^{3} x^{3} - 3\,n^{3} x^{4} - 18\,n^{4} x^{2} + 9\,n^{4} x^{3}\right)\,P_{n}(x) = \left(256\,x + 208\,x^{2} + 144\,x^{3} + 40\,x^{4} - 8\,x^{5} - 64\,n + 64\,n x + 616\,n x^{2} + 392\,n x^{3} + 106\,n x^{4} - 14\,n x^{5} - 64\,n^{2} - 320\,n^{2} x + 284\,n^{2} x^{3} + 121\,n^{2} x^{4} - 6\,n^{2} x^{5} - 144\,n^{3} x - 276\,n^{3} x^{2} - 48\,n^{3} x^{3} + 39\,n^{3} x^{4} - 36\,n^{4} x^{2} - 36\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(64\,x + 192\,x^{2} + 176\,x^{3} + 64\,x^{4} - 24\,x^{5} - 4\,x^{6} + 64\,n x + 384\,n x^{2} + 632\,n x^{3} + 244\,n x^{4} - 30\,n x^{5} - 7\,n x^{6} + 144\,n^{2} x^{2} + 420\,n^{2} x^{3} + 288\,n^{2} x^{4} + 9\,n^{2} x^{5} - 3\,n^{2} x^{6} + 36\,n^{3} x^{3} + 72\,n^{3} x^{4} + 9\,n^{3} x^{5}\right)\,P_{n-1}'(x)\)

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

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	4														6
2	4	20	9													33
3	8	64	72	16												160
4	16	176	333	176	25											726
5	32	448	1206	1072	350	36										3144
6	64	1088	3816	4880	2650	612	49									13159
7	128	2560	11088	18688	14600	5544	980	64								53652
8	256	5888	30384	63872	66025	35928	10339	1472	81							214245
9	512	13312	79776	201472	261050	187956	77126	17728	2106	100						841138
10	1024	29696	202752	598784	936700	846900	458983	149696	28512	2900	121					3256068
11	2048	65536	502272	1699840	3123800	3418560	2326520	999616	268920	43600	3872	144				12454728
12	4096	143360	1218816	4653056	9839200	12684240	10458805	5637440	1992438	454400	64009	5040	169			47155069
13	8192	311296	2907648	12365824	29604800	44035200	42835310	27974080	12384900	3700600	730598	90864	6422	196		176955930
14	16384	671744	6838272	32067584	85801600	144869760	162872080	125556800	67309380	25151000	6488746	1127376	125398	8036	225	658904385

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Technical details

Node id: n-4b7add9985ac0be4 · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-e6fcc40b70107e9e

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

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;
  1:   2,     4;
  2:   4,    20,     9;
  3:   8,    64,    72,     16;
  4:  16,   176,   333,    176,     25;
  5:  32,   448,  1206,   1072,    350,     36;
  6:  64,  1088,  3816,   4880,   2650,    612,    49;
  7: 128,  2560, 11088,  18688,  14600,   5544,   980,    64;
  8: 256,  5888, 30384,  63872,  66025,  35928, 10339,  1472,   81;
  9: 512, 13312, 79776, 201472, 261050, 187956, 77126, 17728, 2106, 100;

%F Formula

The row polynomials satisfy -(32*n^2+32*n+(72*n^3-32*n^2-96*n)*x+(18*n^4-42*n^3-120*n^2)*x^2-(9*n^4+39*n^3+16*n^2+16*n)*x^3+(3*n^3+4*n^2)*x^4)*P(n,x) = -(64*n^2+64*n+(144*n^3+320*n^2-64*n-256)*x+(36*n^4+276*n^3-616*n-208)*x^2+(36*n^4+48*n^3-284*n^2-392*n-144)*x^3-(39*n^3+121*n^2+106*n+40)*x^4+(6*n^2+14*n+8)*x^5)*P(n-1,x) + ((64*n+64)*x+(144*n^2+384*n+192)*x^2+(36*n^3+420*n^2+632*n+176)*x^3+(72*n^3+288*n^2+244*n+64)*x^4+(9*n^3+9*n^2-30*n-24)*x^5-(3*n^2+7*n+4)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(64*n^2 + 64*n + (144*n^3 + 320*n^2 - 64*n - 256)*x + (36*n^4 + 276*n^3 - 616*n - 208)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (39*n^3 + 121*n^2 + 106*n + 40)*x^4 + (6*n^2 + 14*n + 8)*x^5)*P[n - 1, x] + ((64*n + 64)*x + (144*n^2 + 384*n + 192)*x^2 + (36*n^3 + 420*n^2 + 632*n + 176)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (9*n^3 + 9*n^2 - 30*n - 24)*x^5 - (3*n^2 + 7*n + 4)*x^6)*D[P[n - 1, x], x])/(-32*n^2 - 32*n - (72*n^3 - 32*n^2 - 96*n)*x - (18*n^4 - 42*n^3 - 120*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (3*n^3 + 4*n^2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(64*n^2 + 64*n + (144*n^3 + 320*n^2 - 64*n - 256)*x + (36*n^4 + 276*n^3 - 616*n - 208)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (39*n^3 + 121*n^2 + 106*n + 40)*x^4 + (6*n^2 + 14*n + 8)*x^5)*v[i - 1] + ((64*n + 64)*x + (144*n^2 + 384*n + 192)*x^2 + (36*n^3 + 420*n^2 + 632*n + 176)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (9*n^3 + 9*n^2 - 30*n - 24)*x^5 - (3*n^2 + 7*n + 4)*x^6)*deriv(v[i - 1], x))/(-32*n^2 - 32*n - (72*n^3 - 32*n^2 - 96*n)*x - (18*n^4 - 42*n^3 - 120*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (3*n^3 + 4*n^2)*x^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)

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-01576e9f8151fd70
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-01576e9f8151fd70
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-01576e9f8151fd70
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-01576e9f8151fd70

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-01576e9f8151fd70

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-81aa6611c032ec70 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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
[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.