A357613 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 6\,n x - 6\,n x^{2} - 2\,n x^{3} + 4\,n^{2} + 12\,n^{2} x + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n}(x) = \left(6 + 33\,x + 69\,x^{2} + 69\,x^{3} + 33\,x^{4} + 6\,x^{5} - 27\,n - 146\,n x - 302\,n x^{2} - 300\,n x^{3} - 143\,n x^{4} - 26\,n x^{5} + 27\,n^{2} + 141\,n^{2} x + 281\,n^{2} x^{2} + 267\,n^{2} x^{3} + 120\,n^{2} x^{4} + 20\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(3\,x + 18\,x^{2} + 42\,x^{3} + 48\,x^{4} + 27\,x^{5} + 6\,x^{6} - 9\,n x - 55\,n x^{2} - 131\,n x^{3} - 153\,n x^{4} - 88\,n x^{5} - 20\,n x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	5														8
2	15	50	41													106
3	84	420	693	377												1574
4	495	3300	8190	8964	3649											24598
5	3003	25025	82940	136620	111815	36365										395768
6	18564	185640	769860	1694420	2087085	1363830	369305									6488704
7	116280	1356600	6755868	18613980	30640120	30128280	16383465	3800305								107794898
8	735471	9806280	57004332	188675928	388869390	511004088	418054140	194654216	39490049							1808293894
9	4686825	70302375	467240400	1805739936	4471750206	7358114610	8044270920	5633882280	2293485003	413442773						30562915328
10	30045015	500750250	3745266525	16552879200	47871723900	94655345400	129579925860	121264549560	74238771915	26846417850	4354393801					519640069276
11	193536720	3548173200	29494189725	146726976825	485355756600	1120865545800	1843914372300	2160720061500	1767377419500	960992806620	312600525853	46082942185				8877872306828
12	1251677700	25033554000	228949732440	1266080758800	4714712273700	12454782139800	23931742734900	33700004137800	34514678323125	25071880333500	12260994195450	3624203551500	489658242241			152283971654956
13	8122425444	175985884620	1756153880208	10686523799952	44241716539020	131582131846308	289233627059520	475736199434496	585507739545729	532520640046395	347878156229604	154569598010100	41866048057743	5220495115997		2620983137875136
14	52860229080	1233405345200	13335819256760	88556097140784	403473060674684	1334193245064680	3302030663191260	6213367393559040	8932187322386328	9762112947835248	7984198554932600	4738512178132460	1929020476051305	482142406433534	55818956905529	45240235387138492

Technical details

Node id: n-68a79d6800b9e1df · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-bad64ddddd969224

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:       3,        5;
  2:      15,       50,        41;
  3:      84,      420,       693,        377;
  4:     495,     3300,      8190,       8964,       3649;
  5:    3003,    25025,     82940,     136620,     111815,      36365;
  6:   18564,   185640,    769860,    1694420,    2087085,    1363830,     369305;
  7:  116280,  1356600,   6755868,   18613980,   30640120,   30128280,   16383465,    3800305;
  8:  735471,  9806280,  57004332,  188675928,  388869390,  511004088,  418054140,  194654216,   39490049;
  9: 4686825, 70302375, 467240400, 1805739936, 4471750206, 7358114610, 8044270920, 5633882280, 2293485003, 413442773;

%F Formula

The row polynomials satisfy (4*n^2-2*n+(12*n^2-6*n)*x+(12*n^2-6*n)*x^2+(4*n^2-2*n)*x^3)*P(n,x) = (27*n^2-27*n+6+(141*n^2-146*n+33)*x+(281*n^2-302*n+69)*x^2+(267*n^2-300*n+69)*x^3+(120*n^2-143*n+33)*x^4+(20*n^2-26*n+6)*x^5)*P(n-1,x) - ((9*n-3)*x+(55*n-18)*x^2+(131*n-42)*x^3+(153*n-48)*x^4+(88*n-27)*x^5+(20*n-6)*x^6)*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[((27*n^2 - 27*n + 6 + (141*n^2 - 146*n + 33)*x + (281*n^2 - 302*n + 69)*x^2 + (267*n^2 - 300*n + 69)*x^3 + (120*n^2 - 143*n + 33)*x^4 + (20*n^2 - 26*n + 6)*x^5)*P[n - 1, x] - ((9*n - 3)*x + (55*n - 18)*x^2 + (131*n - 42)*x^3 + (153*n - 48)*x^4 + (88*n - 27)*x^5 + (20*n - 6)*x^6)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (12*n^2 - 6*n)*x + (12*n^2 - 6*n)*x^2 + (4*n^2 - 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] = ((27*n^2 - 27*n + 6 + (141*n^2 - 146*n + 33)*x + (281*n^2 - 302*n + 69)*x^2 + (267*n^2 - 300*n + 69)*x^3 + (120*n^2 - 143*n + 33)*x^4 + (20*n^2 - 26*n + 6)*x^5)*v[i - 1] - ((9*n - 3)*x + (55*n - 18)*x^2 + (131*n - 42)*x^3 + (153*n - 48)*x^4 + (88*n - 27)*x^5 + (20*n - 6)*x^6)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (12*n^2 - 6*n)*x + (12*n^2 - 6*n)*x^2 + (4*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 - 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) = binomial(2*n, k) * binomial(3*n - k, 2*n).?
  - 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

References

[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.