A359363 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(6 + 18\,x + 18\,x^{2} + 6\,x^{3} + 5\,n + 15\,n x + 15\,n x^{2} + 5\,n x^{3} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n}(x) = \left(-3 - 6\,x + 24\,x^{2} + 102\,x^{3} + 147\,x^{4} + 96\,x^{5} + 24\,x^{6} + 2\,n + 7\,n x - 33\,n x^{2} - 157\,n x^{3} - 233\,n x^{4} - 150\,n x^{5} - 36\,n x^{6} + n^{2} + 11\,n^{2} x + 45\,n^{2} x^{2} + 91\,n^{2} x^{3} + 98\,n^{2} x^{4} + 54\,n^{2} x^{5} + 12\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(9\,x + 72\,x^{2} + 246\,x^{3} + 444\,x^{4} + 441\,x^{5} + 228\,x^{6} + 48\,x^{7} + 3\,n x - 12\,n x^{2} - 102\,n x^{3} - 228\,n x^{4} - 237\,n x^{5} - 120\,n x^{6} - 24\,n x^{7}\right)\,P_{n-1}'(x) + \left(6\,x^{2} + 42\,x^{3} + 120\,x^{4} + 180\,x^{5} + 150\,x^{6} + 66\,x^{7} + 12\,x^{8}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	1	2													3
3	0	1	6	6												13
4	0	1	13	33	22											69
5	0	1	24	116	184	92										417
6	0	1	40	325	920	1055	422									2763
7	0	1	62	785	3520	6945	6222	2074								19609
8	0	1	91	1701	11291	34055	51261	37639	10754							146793
9	0	1	128	3388	31808	137326	308672	374500	232808	58202						1146833
10	0	1	174	6306	80934	479220	1507170	2685690	2724780	1468080	326240					9278595
11	0	1	230	11100	189540	1492470	6299160	15451800	22742400	19807800	9414800	1882960				77292261
12	0	1	297	18645	414150	4234560	23287440	75453840	151021200	189006840	144123320	61273080	11140560			659973933
13	0	1	376	30096	853072	11108592	77841984	323657400	845463696	1425339696	1549945760	1050619064	403979952	67329992		5756169681
14	0	1	468	46943	1669954	27245933	238989036	1247920245	4134563004	9012369795	13101963446	12586870725	7677014644	2694246347	414499438	51137399979

Technical details

Node id: n-6a437466bd6a8bc0 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-bb0a136ce054ce63

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: 0, 1;
  2: 0, 1,   2;
  3: 0, 1,   6,    6;
  4: 0, 1,  13,   33,    22;
  5: 0, 1,  24,  116,   184,     92;
  6: 0, 1,  40,  325,   920,   1055,    422;
  7: 0, 1,  62,  785,  3520,   6945,   6222,   2074;
  8: 0, 1,  91, 1701, 11291,  34055,  51261,  37639,  10754;
  9: 0, 1, 128, 3388, 31808, 137326, 308672, 374500, 232808, 58202;

%F Formula

The row polynomials satisfy (n^2+5*n+6+(3*n^2+15*n+18)*x+(3*n^2+15*n+18)*x^2+(n^2+5*n+6)*x^3)*P(n,x) = (n^2+2*n-3+(11*n^2+7*n-6)*x+(45*n^2-33*n+24)*x^2+(91*n^2-157*n+102)*x^3+(98*n^2-233*n+147)*x^4+(54*n^2-150*n+96)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) + ((3*n+9)*x-(12*n-72)*x^2-(102*n-246)*x^3-(228*n-444)*x^4-(237*n-441)*x^5-(120*n-228)*x^6-(24*n-48)*x^7)*d/dx P(n-1,x) + (6*x^2+42*x^3+120*x^4+180*x^5+150*x^6+66*x^7+12*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[((n^2 + 2*n - 3 + (11*n^2 + 7*n - 6)*x + (45*n^2 - 33*n + 24)*x^2 + (91*n^2 - 157*n + 102)*x^3 + (98*n^2 - 233*n + 147)*x^4 + (54*n^2 - 150*n + 96)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] + ((3*n + 9)*x - (12*n - 72)*x^2 - (102*n - 246)*x^3 - (228*n - 444)*x^4 - (237*n - 441)*x^5 - (120*n - 228)*x^6 - (24*n - 48)*x^7)*D[P[n - 1, x], x] + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 5*n + 6 + (3*n^2 + 15*n + 18)*x + (3*n^2 + 15*n + 18)*x^2 + (n^2 + 5*n + 6)*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 + 2*n - 3 + (11*n^2 + 7*n - 6)*x + (45*n^2 - 33*n + 24)*x^2 + (91*n^2 - 157*n + 102)*x^3 + (98*n^2 - 233*n + 147)*x^4 + (54*n^2 - 150*n + 96)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] + ((3*n + 9)*x - (12*n - 72)*x^2 - (102*n - 246)*x^3 - (228*n - 444)*x^4 - (237*n - 441)*x^5 - (120*n - 228)*x^6 - (24*n - 48)*x^7)*deriv(v[i - 1], x) + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 5*n + 6 + (3*n^2 + 15*n + 18)*x + (3*n^2 + 15*n + 18)*x^2 + (n^2 + 5*n + 6)*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. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 …?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.