A272866 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - x^{2} + 3\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 11\,x^{2} + 6\,x^{3} + x^{4} + 3\,n x + 11\,n x^{2} + 9\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	6	3													10
2	1	12	33	24	5											75
3	1	18	90	180	150	54	7									500
4	1	24	174	576	975	864	406	96	9							3125
5	1	30	285	1320	3425	5238	4795	2640	855	150	11					18750
6	1	36	423	2520	8850	19548	27923	26064	15930	6300	1551	216	13			109375
7	1	42	588	4284	19005	55188	108731	147864	139797	91980	41811	12852	2548	294	15	625000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8e8e81fc9d22b8ab · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7820744f6c385cb4

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A053464: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 1,  6,   3;
  2: 1, 12,  33,   24,     5;
  3: 1, 18,  90,  180,   150,     54,      7;
  4: 1, 24, 174,  576,   975,    864,    406,      96,       9;
  5: 1, 30, 285, 1320,  3425,   5238,   4795,    2640,     855,     150,      11;
  6: 1, 36, 423, 2520,  8850,  19548,  27923,   26064,   15930,    6300,    1551,     216,      13;
  7: 1, 42, 588, 4284, 19005,  55188, 108731,  147864,  139797,   91980,   41811,   12852,    2548,    294,     15;
  8: 1, 48, 780, 6720, 36050, 130032, 328496,  594240,  778635,  742800,  516208,  260064,   93730,  23520,   3900,   384,   17;
  9: 1, 54, 999, 9936, 62550, 269892, 834078, 1893888, 3206547, 4081050, 3919113, 2840832, 1549002, 629748, 187650, 39744, 5661, 486, 19;

%F Formula

The row polynomials satisfy (1+3*n*x+(2*n-1)*x^2)*P(n,x) = (1+(3*n+6)*x+(11*n+11)*x^2+(9*n+6)*x^3+(2*n+1)*x^4)*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[((1 + (3*n + 6)*x + (11*n + 11)*x^2 + (9*n + 6)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(1 + 3*n*x + (2*n - 1)*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; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (3*n + 6)*x + (11*n + 11)*x^2 + (9*n + 6)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(1 + 3*n*x + (2*n - 1)*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-c9b2fef810d9f0c1
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-c9b2fef810d9f0c1
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-c9b2fef810d9f0c1

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-c9b2fef810d9f0c1

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-0e058c428dfdfeb6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)

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.