A109983 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(2\,x + 14\,x^{2} + 38\,x^{3} + 50\,x^{4} + 32\,x^{5} + 8\,x^{6} - n x - 8\,n x^{2} - 26\,n x^{3} - 40\,n x^{4} - 29\,n x^{5} - 8\,n x^{6}\right)\,P_{n-1}(x) + \left(2\,x^{2} + 12\,x^{3} + 28\,x^{4} + 32\,x^{5} + 18\,x^{6} + 4\,x^{7}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	0	1	3																											4
2	0	0	1	8	13																									22
3	0	0	0	1	15	57	63																							136
4	0	0	0	0	1	24	156	384	321																					886
5	0	0	0	0	0	1	35	340	1380	2505	1683																			5944
6	0	0	0	0	0	0	1	48	645	3800	11145	16008	8989																	40636
7	0	0	0	0	0	0	0	1	63	1113	8855	37065	84903	100849	48639															281488
8	0	0	0	0	0	0	0	0	1	80	1792	18368	102340	330848	621040	628736	265729													1968934
9	0	0	0	0	0	0	0	0	0	1	99	2736	34944	247716	1055628	2774352	4408848	3888657	1462563											13875544
10	0	0	0	0	0	0	0	0	0	0	1	120	4005	62160	543060	2919168	10041780	22209120	30589065	23900040	8097453									98365972
11	0	0	0	0	0	0	0	0	0	0	0	1	143	5665	104775	1101540	7241388	31222884	89899260	171521625	208411335	146146473	45046719							700701808
12	0	0	0	0	0	0	0	0	0	0	0	0	1	168	7788	168960	2098305	16484688	86427264	310394304	767708865	1287357720	1399199868	889928064	251595969					5011371964
13	0	0	0	0	0	0	0	0	0	0	0	0	0	1	195	10452	262548	3794505	34992243	218103600	948220416	2913982929	6312758595	9439841268	9279936900	5399971161	1409933619			35961808432
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	224	13741	395304	6567561	70078008	510266757	2626557648	9736383657	26124706608	50326289013	67892966232	60919688193	32668236552	7923848253	258805997752

Technical details

Node id: n-d6f28c09f7011372 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-c880512e8efe320e

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1, 3;
  2: 0, 0, 1, 8, 13;
  3: 0, 0, 0, 1, 15, 57,  63;
  4: 0, 0, 0, 0,  1, 24, 156, 384,  321;
  5: 0, 0, 0, 0,  0,  1,  35, 340, 1380, 2505,  1683;
  6: 0, 0, 0, 0,  0,  0,   1,  48,  645, 3800, 11145, 16008,   8989;
  7: 0, 0, 0, 0,  0,  0,   0,   1,   63, 1113,  8855, 37065,  84903, 100849,   48639;
  8: 0, 0, 0, 0,  0,  0,   0,   0,    1,   80,  1792, 18368, 102340, 330848,  621040,  628736,  265729;
  9: 0, 0, 0, 0,  0,  0,   0,   0,    0,    1,    99,  2736,  34944, 247716, 1055628, 2774352, 4408848, 3888657, 1462563;

%F Formula

The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = -((n-2)*x+(8*n-14)*x^2+(26*n-38)*x^3+(40*n-50)*x^4+(29*n-32)*x^5+(8*n-8)*x^6)*P(n-1,x) + (2*x^2+12*x^3+28*x^4+32*x^5+18*x^6+4*x^7)*d/dx 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[(-((n - 2)*x + (8*n - 14)*x^2 + (26*n - 38)*x^3 + (40*n - 50)*x^4 + (29*n - 32)*x^5 + (8*n - 8)*x^6)*P[n - 1, x] + (2*x^2 + 12*x^3 + 28*x^4 + 32*x^5 + 18*x^6 + 4*x^7)*D[P[n - 1, x], x])/(n + 3*n*x + 3*n*x^2 + n*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] = (-((n - 2)*x + (8*n - 14)*x^2 + (26*n - 38)*x^3 + (40*n - 50)*x^4 + (29*n - 32)*x^5 + (8*n - 8)*x^6)*v[i - 1] + (2*x^2 + 12*x^3 + 28*x^4 + 32*x^5 + 18*x^6 + 4*x^7)*deriv(v[i - 1], x))/(n + 3*n*x + 3*n*x^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 - 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: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.?
  - 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

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.