A191935 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 4\,x + 6\,x^{2} + 6\,x^{3} + 7\,x^{4} + 6\,x^{5} + 2\,x^{6} + n x + n x^{2} - 6\,n x^{3} - 14\,n x^{4} - 11\,n x^{5} - 3\,n x^{6} + n^{2} x + 5\,n^{2} x^{2} + 10\,n^{2} x^{3} + 10\,n^{2} x^{4} + 5\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-1}(x) + \left(4\,x^{3} + 16\,x^{4} + 24\,x^{5} + 16\,x^{6} + 4\,x^{7} - 2\,n x^{2} - 10\,n x^{3} - 20\,n x^{4} - 20\,n x^{5} - 10\,n x^{6} - 2\,n x^{7}\right)\,P_{n-1}'(x) + \left(x^{3} + 5\,x^{4} + 10\,x^{5} + 10\,x^{6} + 5\,x^{7} + x^{8}\right)\,P_{n-1}''(x)\)

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

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	3														4
2	1	10	13													24
3	1	23	95	81												200
4	1	44	418	1028	669											2160
5	1	75	1382	7530	13145	6955										29088
6	1	118	3767	39572	145103	197622	88505									474688
7	1	175	8925	163459	1129155	3078893	3450527	1346209								9177344
8	1	248	19012	561976	6737134	33508008	72537364	69221544	23998521							206583808
9	1	339	37260	1673492	32509526	276648082	1056930412	1897245076	1580045745	493956467						5339046400
10	1	450	68289	4442360	132090674	1816695948	11633797274	35808897528	54930338605	40662800066	11596542533					156585673728
11	1	583	118459	10737749	466248970	9847232870	101033918774	509421090122	1308602996613	1753776309891	1170253591423	307301505073				5160713750528
12	1	740	196262	24015068	1464759219	45438172584	715147390948	5700629615224	23458108902935	51631961219124	61488870620182	37396010343404	9110471500693			189548126736384
13	1	923	312754	50313406	4173690807	183071637333	4244788402012	51635132744932	330565966445727	1142325535851989	2198097004643666	2357415704499966	1318540758661449	299893197116059		7702905384321024
14	1	1134	482027	99703916	10948062697	657454669170	21660099951115	388167507605864	3759436845622843	19872294356477042	58993200585146905	100821793565011052	98427179458832859	51007248989607790	10888674034993905	344180323946168320

Technical details

Node id: n-9c17ab385b1aecce · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-cb8f91bdd594c2ea

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,   3;
  2: 1,  10,    13;
  3: 1,  23,    95,      81;
  4: 1,  44,   418,    1028,      669;
  5: 1,  75,  1382,    7530,    13145,      6955;
  6: 1, 118,  3767,   39572,   145103,    197622,      88505;
  7: 1, 175,  8925,  163459,  1129155,   3078893,    3450527,    1346209;
  8: 1, 248, 19012,  561976,  6737134,  33508008,   72537364,   69221544,   23998521;
  9: 1, 339, 37260, 1673492, 32509526, 276648082, 1056930412, 1897245076, 1580045745, 493956467;

%F Formula

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+(n^2+n+4)*x+(5*n^2+n+6)*x^2+(10*n^2-6*n+6)*x^3+(10*n^2-14*n+7)*x^4+(5*n^2-11*n+6)*x^5+(n^2-3*n+2)*x^6)*P(n-1,x) - (2*n*x^2+(10*n-4)*x^3+(20*n-16)*x^4+(20*n-24)*x^5+(10*n-16)*x^6+(2*n-4)*x^7)*d/dx P(n-1,x) + (x^3+5*x^4+10*x^5+10*x^6+5*x^7+x^8)*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[((1 + (n^2 + n + 4)*x + (5*n^2 + n + 6)*x^2 + (10*n^2 - 6*n + 6)*x^3 + (10*n^2 - 14*n + 7)*x^4 + (5*n^2 - 11*n + 6)*x^5 + (n^2 - 3*n + 2)*x^6)*P[n - 1, x] - (2*n*x^2 + (10*n - 4)*x^3 + (20*n - 16)*x^4 + (20*n - 24)*x^5 + (10*n - 16)*x^6 + (2*n - 4)*x^7)*D[P[n - 1, x], x] + (x^3 + 5*x^4 + 10*x^5 + 10*x^6 + 5*x^7 + x^8)*D[P[n - 1, x], {x, 2}])/(1 + 3*x + 3*x^2 + x^3)]];
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 + (n^2 + n + 4)*x + (5*n^2 + n + 6)*x^2 + (10*n^2 - 6*n + 6)*x^3 + (10*n^2 - 14*n + 7)*x^4 + (5*n^2 - 11*n + 6)*x^5 + (n^2 - 3*n + 2)*x^6)*v[i - 1] - (2*n*x^2 + (10*n - 4)*x^3 + (20*n - 16)*x^4 + (20*n - 24)*x^5 + (10*n - 16)*x^6 + (2*n - 4)*x^7)*deriv(v[i - 1], x) + (x^3 + 5*x^4 + 10*x^5 + 10*x^6 + 5*x^7 + x^8)*deriv(deriv(v[i - 1], x), x))/(1 + 3*x + 3*x^2 + x^3)); 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 Triangle read by rows of Legendre-Stirling numbers of the second kind.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A071951 → x ↦ x+1

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.