A080248 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(2 + 4\,x + 2\,x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x + 24\,x^{2} + 22\,x^{3} + 10\,x^{4} + 2\,x^{5} - 4\,n x - 15\,n x^{2} - 21\,n x^{3} - 13\,n x^{4} - 3\,n x^{5} + n^{2} x^{2} + 3\,n^{2} x^{3} + 3\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(4\,x + 20\,x^{2} + 40\,x^{3} + 40\,x^{4} + 20\,x^{5} + 4\,x^{6} - 2\,n x^{2} - 8\,n x^{3} - 12\,n x^{4} - 8\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)

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

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	2														3
2	1	6	6													13
3	1	16	39	25												81
4	1	44	199	290	135											669
5	1	126	972	2443	2504	909										6955
6	1	370	4824	18824	32011	25058	7417									88505
7	1	1100	24877	142989	360329	457318	287897	71698								1346209
8	1	3288	133869	1103210	3885764	7129944	7172659	3762818	806968							23998521
9	1	9850	747330	8756355	41705747	104102597	149052744	123681607	55472411	10427825						493956467
10	1	29534	4286998	71909056	454259950	1486622684	2846091635	3324601908	2340716986	915108084	152915697					11596542533
11	1	88584	25042731	612181461	5071997138	21247186524	52335102114	80795725933	79446127780	48475690274	16772482772	2519879761				307301505073
12	1	265732	147953235	5396904226	58330680981	307874352200	951068269722	1865797914084	2404526210473	2036780252700	1094886237806	339386061404	46276398129			9110471500693
13	1	797174	880013224	49092431675	692219852466	4556865362349	17344978975332	42111204988174	68357404653987	75413168214228	56012525492054	26875970697008	7538589570965	940296067422		299893197116059
14	1	2391498	5254569828	458387742760	8478425710613	69192756784566	320475191070417	945016631783304	1880710690639609	2596202197109778	2499156589709430	1650587659019504	714503765946825	182879382665542	21007099850230	10888674034993905

Technical details

Node id: n-280ad20172ff23e1 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-82391858922dbeb5

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,    2;
  2: 1,    6,      6;
  3: 1,   16,     39,      25;
  4: 1,   44,    199,     290,      135;
  5: 1,  126,    972,    2443,     2504,       909;
  6: 1,  370,   4824,   18824,    32011,     25058,      7417;
  7: 1, 1100,  24877,  142989,   360329,    457318,    287897,     71698;
  8: 1, 3288, 133869, 1103210,  3885764,   7129944,   7172659,   3762818,   806968;
  9: 1, 9850, 747330, 8756355, 41705747, 104102597, 149052744, 123681607, 55472411, 10427825;

%F Formula

The row polynomials satisfy (2+4*x+2*x^2)*P(n,x) = (2-(4*n-12)*x+(n^2-15*n+24)*x^2+(3*n^2-21*n+22)*x^3+(3*n^2-13*n+10)*x^4+(n^2-3*n+2)*x^5)*P(n-1,x) + (4*x-(2*n-20)*x^2-(8*n-40)*x^3-(12*n-40)*x^4-(8*n-20)*x^5-(2*n-4)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+x^7)*d^2/dx^2 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[((2 - (4*n - 12)*x + (n^2 - 15*n + 24)*x^2 + (3*n^2 - 21*n + 22)*x^3 + (3*n^2 - 13*n + 10)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] + (4*x - (2*n - 20)*x^2 - (8*n - 40)*x^3 - (12*n - 40)*x^4 - (8*n - 20)*x^5 - (2*n - 4)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(2 + 4*x + 2*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] = ((2 - (4*n - 12)*x + (n^2 - 15*n + 24)*x^2 + (3*n^2 - 21*n + 22)*x^3 + (3*n^2 - 13*n + 10)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] + (4*x - (2*n - 20)*x^2 - (8*n - 40)*x^3 - (12*n - 40)*x^4 - (8*n - 20)*x^5 - (2*n - 4)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), x))/(2 + 4*x + 2*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Stirling-like number triangle defined by sequence A000217.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.