A371081 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(9 + 35\,x + 52\,x^{2} + 37\,x^{3} + 13\,x^{4} + 2\,x^{5} + 6\,n + 13\,n x - 18\,n x^{3} - 14\,n x^{4} - 3\,n x^{5} + n^{2} + n^{2} x - 2\,n^{2} x^{2} - 2\,n^{2} x^{3} + n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}(x) + \left(7\,x + 32\,x^{2} + 58\,x^{3} + 52\,x^{4} + 23\,x^{5} + 4\,x^{6} + 2\,n x + 6\,n x^{2} + 4\,n x^{3} - 4\,n x^{4} - 6\,n x^{5} - 2\,n x^{6}\right)\,P_{n-1}'(x) + \left(x^{2} + 5\,x^{3} + 10\,x^{4} + 10\,x^{5} + 5\,x^{6} + x^{7}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	16	17														33
2	400	852	453													1705
3	14400	46148	49212	17465												127225
4	705600	3025472	4855160	3456520	921233											12963985
5	45158400	242946432	521639200	558861488	298799464	63789145										1731194129
6	3657830400	23707584000	63861552544	91533956768	73641038360	31535479548	5616599013									293554040633
7	365783040000	2777171904000	9010864935936	16200522990272	17434500727040	11232879103268	4012573451636	613148157073								61647444309225
8	44259747840000	385647215616000	1465422132893184	3172785038051072	4282070715283392	3689768554863360	1982718348877776	607568974811920	81298838448001							15711539566684705
9	6373403688960000	62740093307904000	273522037261400064	693369747801522432	1126654236733427968	1217238635615835840	874615142613135424	403092467127352960	108147204453430848	12871080897739073						4778624049500708609
10	1077105223434240000	11831799819055104000	58256703087226214400	169380982415956359168	322158978240748119040	418952363098198062592	377356990727709169664	232506327082074248000	93803996974920596224	22380631216915195428	2398329378160629861					1710104207264397938377
11	211112623793111040000	2561984173508014080000	14071209555233142374400	46191063904165479370752	100736148503549341624320	153302213779692083952128	166167318367729272547840	128317002850699599195968	69196399583902975659328	24821938376479151421860	5331568828289995211948	519554377953510437129				711427514924995676915673
12	47500340353449984000000	631589996681843834880000	3830787700700084468121600	14022470653044105591963648	34516565116696178898352128	60213674479669796187392000	76358279810334251135721472	70943050709948733093769984	47938020581031606250289792	22980925831427301536504128	7420229090396254014880776	1449143149124739619826712	129472180384695112970705			340481709639793039728672945
13	12160087130483195904000000	175926961731923320504320000	1168634075488741407581798400	4722309255056566660929404928	12957727567300935891615203328	25506441279916903320192663552	37068647705113033754381068288	40283489941085252606557629184	32743486987774992630126376192	19665456977686155360586588352	8484318010662140452647935072	2490436316080762174487966832	445859544668153603623487384	36773246580917492621295817		185761667956276961871759737329
14	3514265180709643616256000000	54993921224613428519239680000	397305726753322297421719142400	1757632980496876292627457638400	5322285162751908997541916573696	11675203382366992904348673589248	19140943445249703964716525463552	23833086630543754060865514055680	22654565840994402779908216739328	16363724448548604019086210056704	8843290580502196942770138721568	3467920092049695681029859165408	933029565193062132084796017992	154180986060917494169592182220	11807854666147122586977709125	114613484882582907761293092735321

Technical details

Node id: n-50b777e2c50736e8 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-9558912bc02a9014

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:               16,                17;
  2:              400,               852,                453;
  3:            14400,             46148,              49212,              17465;
  4:           705600,           3025472,            4855160,            3456520,              921233;
  5:         45158400,         242946432,          521639200,          558861488,           298799464,            63789145;
  6:       3657830400,       23707584000,        63861552544,        91533956768,         73641038360,         31535479548,         5616599013;
  7:     365783040000,     2777171904000,      9010864935936,     16200522990272,      17434500727040,      11232879103268,      4012573451636,       613148157073;
  8:   44259747840000,   385647215616000,   1465422132893184,   3172785038051072,    4282070715283392,    3689768554863360,   1982718348877776,    607568974811920,     81298838448001;
  9: 6373403688960000, 62740093307904000, 273522037261400064, 693369747801522432, 1126654236733427968, 1217238635615835840, 874615142613135424, 403092467127352960, 108147204453430848, 12871080897739073;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n^2+6*n+9+(n^2+13*n+35)*x-(2*n^2-52)*x^2-(2*n^2+18*n-37)*x^3+(n^2-14*n+13)*x^4+(n^2-3*n+2)*x^5)*P(n-1,x) + ((2*n+7)*x+(6*n+32)*x^2+(4*n+58)*x^3-(4*n-52)*x^4-(6*n-23)*x^5-(2*n-4)*x^6)*d/dx P(n-1,x) + (x^2+5*x^3+10*x^4+10*x^5+5*x^6+x^7)*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 + 6*n + 9 + (n^2 + 13*n + 35)*x - (2*n^2 - 52)*x^2 - (2*n^2 + 18*n - 37)*x^3 + (n^2 - 14*n + 13)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] + ((2*n + 7)*x + (6*n + 32)*x^2 + (4*n + 58)*x^3 - (4*n - 52)*x^4 - (6*n - 23)*x^5 - (2*n - 4)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + x^2)]];
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 + 6*n + 9 + (n^2 + 13*n + 35)*x - (2*n^2 - 52)*x^2 - (2*n^2 + 18*n - 37)*x^3 + (n^2 - 14*n + 13)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] + ((2*n + 7)*x + (6*n + 32)*x^2 + (4*n + 58)*x^3 - (4*n - 52)*x^4 - (6*n - 23)*x^5 - (2*n - 4)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + 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 read by rows, (2, 2)-Lah numbers.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.