A133214 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(11 + 56\,x + 94\,x^{2} + 64\,x^{3} + 15\,x^{4} + 8\,n + 43\,n x + 77\,n x^{2} + 57\,n x^{3} + 15\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 11\,x - 42\,x^{2} - 70\,x^{3} - 53\,x^{4} - 15\,x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 13 + 58\,x + 61\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	13	58	61								132
1	63	417	885	595							1960
2	321	2820	9030	12420	6145						30736
3	1683	18435	78990	164950	167215	65527					496800
4	8989	117966	633315	1776580	2739795	2196366	712909				8185920
5	48639	743869	4800243	16917705	35110285	42830151	28379841	7863667			136694400
6	265729	4640776	34981660	148488760	387744070	636984376	642007324	362435080	87615745		2305163520
7	1462563	28717695	247630860	1229734380	3872306970	8010304722	10874222940	9331525980	4588123995	983726695	39167756800

All 15 rows

Technical details

Node id: n-79e8e9ac32c0a215 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-c57d05a45d068312

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:       13,         58,          61;
  1:       63,        417,         885,         595;
  2:      321,       2820,        9030,       12420,         6145;
  3:     1683,      18435,       78990,      164950,       167215,        65527;
  4:     8989,     117966,      633315,     1776580,      2739795,      2196366,        712909;
  5:    48639,     743869,     4800243,    16917705,     35110285,     42830151,      28379841,       7863667;
  6:   265729,    4640776,    34981660,   148488760,    387744070,    636984376,     642007324,     362435080,      87615745;
  7:  1462563,   28717695,   247630860,  1229734380,   3872306970,   8010304722,   10874222940,    9331525980,    4588123995,     983726695;
  8:  8097453,  176574690,  1714211865,  9750084120,  35955161130,  89756439948,  153485216010,  177380457240,  132475667625,   57685960290,  11112210781;
  9: 45046719, 1080099801, 11658009825, 74727690015, 315893935830, 924117914874, 1907810530626, 2777634711390, 2793027700155, 1846055225245, 721318343053, 126142119187;

%F Formula

The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (8*n+11+(43*n+56)*x+(77*n+94)*x^2+(57*n+64)*x^3+(15*n+15)*x^4)*P(n-1,x) - (1+11*x+42*x^2+70*x^3+53*x^4+15*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 58*x + 61*x^2.

%t Mathematica

P[0, x_] := 13 + 58*x + 61*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 11 + (43*n + 56)*x + (77*n + 94)*x^2 + (57*n + 64)*x^3 + (15*n + 15)*x^4)*P[n - 1, x] - (1 + 11*x + 42*x^2 + 70*x^3 + 53*x^4 + 15*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 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] = 13 + 58*x + 61*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 11 + (43*n + 56)*x + (77*n + 94)*x^2 + (57*n + 64)*x^3 + (15*n + 15)*x^4)*v[i - 1] - (1 + 11*x + 42*x^2 + 70*x^3 + 53*x^4 + 15*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 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 basis_map_A013609 of Delannoy paths counted by number of weak peaks.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Delannoy paths counted by number of weak peaks.?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Delannoy paths counted by number of weak peaks.?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

A104684 → basis A013611 → x ↦ x/(x+1)

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.
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[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