A286785 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(56 + 112\,x + 56\,x^{2} + 30\,n + 60\,n x + 30\,n x^{2} + 4\,n^{2} + 8\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(154 + 518\,x + 574\,x^{2} + 210\,x^{3} + 93\,n + 309\,n x + 339\,n x^{2} + 123\,n x^{3} + 14\,n^{2} + 46\,n^{2} x + 50\,n^{2} x^{2} + 18\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(2 + 36\,x + 96\,x^{2} + 92\,x^{3} + 30\,x^{4} + n + 12\,n x + 30\,n x^{2} + 28\,n x^{3} + 9\,n x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 14 + 42\,x + 30\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	14	42	30															86
1	42	198	297	143														680
2	132	858	2002	2002	728													5722
3	429	3575	11440	17680	13260	3876												50260
4	1430	14586	59670	125970	145350	87210	21318											455534
5	4862	58786	293930	791350	1243550	1144066	572033	120175										4228752
6	16796	235144	1392776	4576264	9152528	11440660	8748740	3749460	690690									40003058
7	58786	936054	6418656	24961440	60843510	96633810	100212840	65552760	24582285	4032015								384232156
8	208012	3714500	28973100	130378950	375798150	726543090	955977750	846723150	483841800	161280600	23841480							3737280582
9	742900	14709420	128707425	658679175	2195597250	5015206350	8024330160	9006901200	6959878200	3530373000	1059111900	142498692						36736735672
10	2674440	58169070	564582150	3241119750	12282138000	32424844320	61246928160	83518538400	81702918000	55981629000	25527622824	6962078952	859515920					364372758986
11	9694845	229824855	2451465120	15611962080	66350838840	199052516520	434296399680	698650729920	829647741780	719028042876	442478795616	183248794144	45812198536	5225264024				3642094268836
12	35357670	907513530	10555815270	73890706890	348341903910	1171695494970	2903767096230	5392710321570	7549794450198	7936963396362	6173193752726	3447367939834	1307622322006	301758997386	31983672534			36650588741086
13	129644790	3582290250	45136857150	344616322050	1785739123350	6661583338410	18504398162250	39017845267830	63028826971110	78072744519770	73611444832926	51920298079650	26537041240710	9284671997370	1989572570865	196947587823		371004578806304
14	477638700	14138105520	191874289200	1587323665200	8971829412000	36784500589200	113296261814736	267677981210640	490742965552840	701061379361200	778419876394160	665195167100464	429158172322880	202199523498280	65649195941000	13129839188200	1219199353190	3775299705437410

Technical details

Node id: n-8f81b8435f04f925 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-caa98baf3ef4c119

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:     14,       42,        30;
  1:     42,      198,       297,       143;
  2:    132,      858,      2002,      2002,        728;
  3:    429,     3575,     11440,     17680,      13260,       3876;
  4:   1430,    14586,     59670,    125970,     145350,      87210,      21318;
  5:   4862,    58786,    293930,    791350,    1243550,    1144066,     572033,     120175;
  6:  16796,   235144,   1392776,   4576264,    9152528,   11440660,    8748740,    3749460,     690690;
  7:  58786,   936054,   6418656,  24961440,   60843510,   96633810,  100212840,   65552760,   24582285,    4032015;
  8: 208012,  3714500,  28973100, 130378950,  375798150,  726543090,  955977750,  846723150,  483841800,  161280600,   23841480;
  9: 742900, 14709420, 128707425, 658679175, 2195597250, 5015206350, 8024330160, 9006901200, 6959878200, 3530373000, 1059111900, 142498692;

%F Formula

The row polynomials satisfy (4*n^2+30*n+56+(8*n^2+60*n+112)*x+(4*n^2+30*n+56)*x^2)*P(n,x) = (14*n^2+93*n+154+(46*n^2+309*n+518)*x+(50*n^2+339*n+574)*x^2+(18*n^2+123*n+210)*x^3)*P(n-1,x) + (n+2+(12*n+36)*x+(30*n+96)*x^2+(28*n+92)*x^3+(9*n+30)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 42*x + 30*x^2.

%t Mathematica

P[0, x_] := 14 + 42*x + 30*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 + 93*n + 154 + (46*n^2 + 309*n + 518)*x + (50*n^2 + 339*n + 574)*x^2 + (18*n^2 + 123*n + 210)*x^3)*P[n - 1, x] + (n + 2 + (12*n + 36)*x + (30*n + 96)*x^2 + (28*n + 92)*x^3 + (9*n + 30)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56 + (8*n^2 + 60*n + 112)*x + (4*n^2 + 30*n + 56)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 14 + 42*x + 30*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 + 93*n + 154 + (46*n^2 + 309*n + 518)*x + (50*n^2 + 339*n + 574)*x^2 + (18*n^2 + 123*n + 210)*x^3)*v[i - 1] + (n + 2 + (12*n + 36)*x + (30*n + 96)*x^2 + (28*n + 92)*x^3 + (9*n + 30)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56 + (8*n^2 + 60*n + 112)*x + (4*n^2 + 30*n + 56)*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 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.?
  - 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.