A064861 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 54\,x + 122\,x^{2} + 108\,x^{3} + 32\,x^{4} + 18\,n x + 81\,n x^{2} + 112\,n x^{3} + 48\,n x^{4} + 9\,n^{2} x^{2} + 24\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n}(x) = \left(8 + 96\,x + 444\,x^{2} + 1024\,x^{3} + 1252\,x^{4} + 776\,x^{5} + 192\,x^{6} + 18\,n x + 153\,n x^{2} + 493\,n x^{3} + 758\,n x^{4} + 560\,n x^{5} + 160\,n x^{6} + 9\,n^{2} x^{2} + 51\,n^{2} x^{3} + 106\,n^{2} x^{4} + 96\,n^{2} x^{5} + 32\,n^{2} x^{6}\right)\,P_{n-2}(x) + \left(6\,x + 26\,x^{2} + 36\,x^{3} + 16\,x^{4}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 6\,x + 6\,x^{2}\)

\(P_{1}(x) = 1 + 10\,x + 24\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	6	6								13
1	1	10	24	16							51
2	1	12	39	48	20						120
3	1	16	75	152	140	48					432
4	1	18	99	252	330	216	56				972
5	1	22	153	516	960	1008	560	128			3348
6	1	24	186	720	1605	2160	1736	768	144		7344
7	1	28	258	1216	3405	6012	6776	4736	1872	320	24624

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ad6cd90b76531a4c · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c7f8d68687ddb27c

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,  6,   6;
  1: 1, 10,  24,   16;
  2: 1, 12,  39,   48,   20;
  3: 1, 16,  75,  152,  140,    48;
  4: 1, 18,  99,  252,  330,   216,    56;
  5: 1, 22, 153,  516,  960,  1008,   560,   128;
  6: 1, 24, 186,  720, 1605,  2160,  1736,   768,   144;
  7: 1, 28, 258, 1216, 3405,  6012,  6776,  4736,  1872,   320;
  8: 1, 30, 300, 1560, 4925, 10098, 13790, 12480,  7200,  2400,  352;
  9: 1, 34, 390, 2360, 8825, 21918, 37352, 44000, 35280, 18400, 5632, 768;

%F Formula

The row polynomials satisfy (8+(18*n+54)*x+(9*n^2+81*n+122)*x^2+(24*n^2+112*n+108)*x^3+(16*n^2+48*n+32)*x^4)*P(n,x) = (8+(18*n+96)*x+(9*n^2+153*n+444)*x^2+(51*n^2+493*n+1024)*x^3+(106*n^2+758*n+1252)*x^4+(96*n^2+560*n+776)*x^5+(32*n^2+160*n+192)*x^6)*P(n-2,x) + (6*x+26*x^2+36*x^3+16*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 1 + 6*x + 6*x^2, P(1,x) = 1 + 10*x + 24*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 1 + 6*x + 6*x^2;
P[1, x_] := 1 + 10*x + 24*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((8 + (18*n + 96)*x + (9*n^2 + 153*n + 444)*x^2 + (51*n^2 + 493*n + 1024)*x^3 + (106*n^2 + 758*n + 1252)*x^4 + (96*n^2 + 560*n + 776)*x^5 + (32*n^2 + 160*n + 192)*x^6)*P[n - 2, x] + (6*x + 26*x^2 + 36*x^3 + 16*x^4)*P[n - 1, x])/(8 + (18*n + 54)*x + (9*n^2 + 81*n + 122)*x^2 + (24*n^2 + 112*n + 108)*x^3 + (16*n^2 + 48*n + 32)*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 + 6*x + 6*x^2; v[2] = 1 + 10*x + 24*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = ((8 + (18*n + 96)*x + (9*n^2 + 153*n + 444)*x^2 + (51*n^2 + 493*n + 1024)*x^3 + (106*n^2 + 758*n + 1252)*x^4 + (96*n^2 + 560*n + 776)*x^5 + (32*n^2 + 160*n + 192)*x^6)*v[i - 2] + (6*x + 26*x^2 + 36*x^3 + 16*x^4)*v[i - 1])/(8 + (18*n + 54)*x + (9*n^2 + 81*n + 122)*x^2 + (24*n^2 + 112*n + 108)*x^3 + (16*n^2 + 48*n + 32)*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-cb26430ba269e72d
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-cb26430ba269e72d
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-cb26430ba269e72d

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-cb26430ba269e72d

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check
    
    Parent-row provers of polytool 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.