A132166 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(48 + 110\,x + 39\,x^{2} - 60\,x^{3} - 37\,x^{4} + 6\,n - 18\,n x - 91\,n x^{2} - 104\,n x^{3} - 37\,n x^{4}\right)\,P_{n-1}(x) + \left(36\,x + 145\,x^{2} + 219\,x^{3} + 147\,x^{4} + 37\,x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 336 + 714\,x + 379\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	336	714	379								1429
1	4536	14721	15897	5713							40867
2	54432	240912	398475	292026	80032						1065877
3	598752	3406032	7701813	8660757	4846548	1080325					26294227
4	6158592	43483392	126556128	194702292	167231238	76114830	14354557				628601029
5	60046272	514942272	1859594760	3679470009	4319414766	3014218107	1159428027	189853438			14796967651
6	560431872	5758376832	25183907424	61661018016	92868160725	88382031048	52022530263	17344782066	2511177679		346292415925
7	5043886848	61571083392	320562481824	945826659648	1755852557589	2137120347960	1711096972029	871096234389	256316427399	33258000301	8097744651379

All 15 rows

Technical details

Node id: n-c0d9843b43e10688 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-935f97559657f6f5

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:          336,           714,            379;
  1:         4536,         14721,          15897,            5713;
  2:        54432,        240912,         398475,          292026,           80032;
  3:       598752,       3406032,        7701813,         8660757,         4846548,         1080325;
  4:      6158592,      43483392,      126556128,       194702292,       167231238,        76114830,         14354557;
  5:     60046272,     514942272,     1859594760,      3679470009,      4319414766,      3014218107,       1159428027,       189853438;
  6:    560431872,    5758376832,    25183907424,     61661018016,     92868160725,     88382031048,      52022530263,     17344782066,      2511177679;
  7:   5043886848,   61571083392,   320562481824,    945826659648,   1755852557589,   2137120347960,    1711096972029,    871096234389,    256316427399,     33258000301;
  8:  44019376128,  635224052736,  3886773351552,  13552599939840,  30159831310128,  45068906035998,   46008114589593,  31781839649040,  14250561002442,   3751683617316,   440953665088;
  9: 374164697088, 6365996582400, 45313782696000, 183914275761216, 480615616038168, 857088503741457, 1070634053880495, 940541706200538, 571073911849818, 228718085727846, 54467017469472, 5849978617585;

%F Formula

The row polynomials satisfy (n+3+(2*n+6)*x+(n+3)*x^2)*P(n,x) = (6*n+48-(18*n-110)*x-(91*n-39)*x^2-(104*n+60)*x^3-(37*n+37)*x^4)*P(n-1,x) + (36*x+145*x^2+219*x^3+147*x^4+37*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 336 + 714*x + 379*x^2.

%t Mathematica

P[0, x_] := 336 + 714*x + 379*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 48 - (18*n - 110)*x - (91*n - 39)*x^2 - (104*n + 60)*x^3 - (37*n + 37)*x^4)*P[n - 1, x] + (36*x + 145*x^2 + 219*x^3 + 147*x^4 + 37*x^5)*D[P[n - 1, x], x])/(n + 3 + (2*n + 6)*x + (n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 336 + 714*x + 379*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 48 - (18*n - 110)*x - (91*n - 39)*x^2 - (104*n + 60)*x^3 - (37*n + 37)*x^4)*v[i - 1] + (36*x + 145*x^2 + 219*x^3 + 147*x^4 + 37*x^5)*deriv(v[i - 1], x))/(n + 3 + (2*n + 6)*x + (n + 3)*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 A convolution triangle of numbers obtained from A036224.?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741