A172431 → ⊙ C(n,k+1) (shift -1) → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(-2\,n - 6\,n x - 6\,n x^{2} - 2\,n x^{3} - 2\,n^{2} - 6\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} + 12\,n^{3} x + 12\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n}(x) = \left(-12\,x - 84\,x^{2} - 216\,x^{3} - 264\,x^{4} - 156\,x^{5} - 36\,x^{6} - 2\,n + 56\,n x + 366\,n x^{2} + 862\,n x^{3} + 980\,n x^{4} + 546\,n x^{5} + 120\,n x^{6} - 2\,n^{2} - 88\,n^{2} x - 411\,n^{2} x^{2} - 806\,n^{2} x^{3} - 802\,n^{2} x^{4} - 402\,n^{2} x^{5} - 81\,n^{2} x^{6} + 4\,n^{3} + 44\,n^{3} x + 147\,n^{3} x^{2} + 232\,n^{3} x^{3} + 194\,n^{3} x^{4} + 84\,n^{3} x^{5} + 15\,n^{3} x^{6}\right)\,P_{n-1}(x) + \left(6\,x + 6\,x^{2} - 72\,x^{3} - 228\,x^{4} - 282\,x^{5} - 162\,x^{6} - 36\,x^{7} - 20\,n x - 16\,n x^{2} + 238\,n x^{3} + 712\,n x^{4} + 848\,n x^{5} + 472\,n x^{6} + 102\,n x^{7} + 8\,n^{2} x - 2\,n^{2} x^{2} - 118\,n^{2} x^{3} - 292\,n^{2} x^{4} - 308\,n^{2} x^{5} - 154\,n^{2} x^{6} - 30\,n^{2} x^{7}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 4\,x^{3} - 10\,x^{4} - 40\,x^{5} - 50\,x^{6} - 28\,x^{7} - 6\,x^{8} - 5\,n x^{2} - 10\,n x^{3} + 25\,n x^{4} + 100\,n x^{5} + 125\,n x^{6} + 70\,n x^{7} + 15\,n x^{8}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	2							3
3	1	6	6						13
4	1	12	31	21					65
5	1	20	96	156	80				353
6	1	30	230	660	780	322			2023
7	1	42	470	2080	4203	3898	1347		12041
8	1	56	861	5425	16485	25578	19509	5798	73713

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-48a9203d1b5d504a · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-727cfd8711a24e2e

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,  2;
   3: 1,  6,    6;
   4: 1, 12,   31,    21;
   5: 1, 20,   96,   156,     80;
   6: 1, 30,  230,   660,    780,    322;
   7: 1, 42,  470,  2080,   4203,   3898,   1347;
   8: 1, 56,  861,  5425,  16485,  25578,  19509,   5798;
   9: 1, 72, 1456, 12376,  52500, 120288, 151116,  97848,  25512;
  10: 1, 90, 2316, 25536, 143892, 453348, 829680, 874548, 491868, 114236;

%F Formula

The row polynomials satisfy (4*n^3-2*n^2-2*n+(12*n^3-6*n^2-6*n)*x+(12*n^3-6*n^2-6*n)*x^2+(4*n^3-2*n^2-2*n)*x^3)*P(n,x) = (4*n^3-2*n^2-2*n+(44*n^3-88*n^2+56*n-12)*x+(147*n^3-411*n^2+366*n-84)*x^2+(232*n^3-806*n^2+862*n-216)*x^3+(194*n^3-802*n^2+980*n-264)*x^4+(84*n^3-402*n^2+546*n-156)*x^5+(15*n^3-81*n^2+120*n-36)*x^6)*P(n-1,x) + ((8*n^2-20*n+6)*x-(2*n^2+16*n-6)*x^2-(118*n^2-238*n+72)*x^3-(292*n^2-712*n+228)*x^4-(308*n^2-848*n+282)*x^5-(154*n^2-472*n+162)*x^6-(30*n^2-102*n+36)*x^7)*d/dx P(n-1,x) - ((5*n-2)*x^2+(10*n-4)*x^3-(25*n-10)*x^4-(100*n-40)*x^5-(125*n-50)*x^6-(70*n-28)*x^7-(15*n-6)*x^8)*d^2/dx^2 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[((4*n^3 - 2*n^2 - 2*n + (44*n^3 - 88*n^2 + 56*n - 12)*x + (147*n^3 - 411*n^2 + 366*n - 84)*x^2 + (232*n^3 - 806*n^2 + 862*n - 216)*x^3 + (194*n^3 - 802*n^2 + 980*n - 264)*x^4 + (84*n^3 - 402*n^2 + 546*n - 156)*x^5 + (15*n^3 - 81*n^2 + 120*n - 36)*x^6)*P[n - 1, x] + ((8*n^2 - 20*n + 6)*x - (2*n^2 + 16*n - 6)*x^2 - (118*n^2 - 238*n + 72)*x^3 - (292*n^2 - 712*n + 228)*x^4 - (308*n^2 - 848*n + 282)*x^5 - (154*n^2 - 472*n + 162)*x^6 - (30*n^2 - 102*n + 36)*x^7)*D[P[n - 1, x], x] - ((5*n - 2)*x^2 + (10*n - 4)*x^3 - (25*n - 10)*x^4 - (100*n - 40)*x^5 - (125*n - 50)*x^6 - (70*n - 28)*x^7 - (15*n - 6)*x^8)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 2*n^2 - 2*n + (12*n^3 - 6*n^2 - 6*n)*x + (12*n^3 - 6*n^2 - 6*n)*x^2 + (4*n^3 - 2*n^2 - 2*n)*x^3)]];
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] = ((4*n^3 - 2*n^2 - 2*n + (44*n^3 - 88*n^2 + 56*n - 12)*x + (147*n^3 - 411*n^2 + 366*n - 84)*x^2 + (232*n^3 - 806*n^2 + 862*n - 216)*x^3 + (194*n^3 - 802*n^2 + 980*n - 264)*x^4 + (84*n^3 - 402*n^2 + 546*n - 156)*x^5 + (15*n^3 - 81*n^2 + 120*n - 36)*x^6)*v[i - 1] + ((8*n^2 - 20*n + 6)*x - (2*n^2 + 16*n - 6)*x^2 - (118*n^2 - 238*n + 72)*x^3 - (292*n^2 - 712*n + 228)*x^4 - (308*n^2 - 848*n + 282)*x^5 - (154*n^2 - 472*n + 162)*x^6 - (30*n^2 - 102*n + 36)*x^7)*deriv(v[i - 1], x) - ((5*n - 2)*x^2 + (10*n - 4)*x^3 - (25*n - 10)*x^4 - (100*n - 40)*x^5 - (125*n - 50)*x^6 - (70*n - 28)*x^7 - (15*n - 6)*x^8)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 2*n^2 - 2*n + (12*n^3 - 6*n^2 - 6*n)*x + (12*n^3 - 6*n^2 - 6*n)*x^2 + (4*n^3 - 2*n^2 - 2*n)*x^3)); 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 [-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-7103e40cdd5e6d87
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-7103e40cdd5e6d87
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-7103e40cdd5e6d87

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 mobius, shift 0 e-7103e40cdd5e6d87

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image hadamard_multiplier, shift -1 e-1491609a6fc5f8c9
    
    Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
    
    premise (sequence): Even row Pascal-square read by antidiagonals.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    theorem_check three_term_recurrence e-83282836b5ac538e 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