A269947 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 - x + 3\,n + 6\,n x + 3\,n x^{2} - 3\,n^{2} - 6\,n^{2} x - 3\,n^{2} x^{2} + n^{3} + 2\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x)\)

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

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	8	25	18												51
4	0	216	899	1186	504											2805
5	0	13824	71576	134339	109346	32760										361845
6	0	1728000	10688824	25810951	30594964	17872596	4127760									90823095
7	0	373248000	2683761984	7894640224	12209488591	10499587924	4769949492	895723920								39326400135
8	0	128024064000	1048927672512	3631075719328	6903610823769	7801422733236	5247950921612	1948095929808	308129028480							27017236892745
9	0	65548320768000	602727313158144	2397210664294592	5397390585784992	7535880792010329	6689080734015412	3689623938848652	1157135274573264	158070191610240						27692667815063625
10	0	47784725839872000	487238485452926976	2187557512876202688	5684661521972311328	9433752224998574001	10377532833264775518	7572764787251918068	3536977090523425416	959941920122347680	115391239875475200					40403602342177828875
11	0	47784725839872000000	535070996018638848000	2675283236814582590976	7874406592361390218688	15124098408492857640328	19820718810488348093001	17960675153349958361518	11117314642562595402068	4500455987736296521416	1076293101917945227680	115506631115350675200				80847608286697835578875
12	0	63601470092869632000000	775828750519517810688000	4273516554897036374125056	14044312445870034392253760	30618884562729365290568984	46526675816872477688701227	50306856084679274239057790	38720765093512959017694484	20798370023570387745559272	7427153494317532064568192	1587361737769234791960960	153854832645647099366400			215297180867476336146544125
13	0	109903340320478724096000000	1450599022688298370596864000	8726044516510325149167648768	31657482029880395320676719104	77192048743305632686309955392	133338147220514714033469493592	167374869801698299808856282603	153890036252000858341906987262	102887786247413536165920184852	48794503008933895819645817064	15584509474540250660146942848	3010409595414685143005639040	266015005644323834804505600		744282354258865694058603040125
14	0	241457638684091756838912000000	3428533594870603755764318208000	22359536454642064171293231415296	88731333866936923197397243863552	239174076590719583926670400586816	462612032581256607376041759373240	660799836545022506125623199794007	705986373469778748757035760176008	564294766067465425592038204121720	333348877282442721608454426394304	141489484929201633711924339343728	40868621706165534210186369350784	7201315258122057409391893413120	584700982406223788900303308800	3271120946967714725387560361349375

Technical details

Node id: n-a97ecc679ffc37ba · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-411590dcfce40f62

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): yes

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,              8,              25,               18;
  4: 0,            216,             899,             1186,              504;
  5: 0,          13824,           71576,           134339,           109346,            32760;
  6: 0,        1728000,        10688824,         25810951,         30594964,         17872596,          4127760;
  7: 0,      373248000,      2683761984,       7894640224,      12209488591,      10499587924,       4769949492,        895723920;
  8: 0,   128024064000,   1048927672512,    3631075719328,    6903610823769,    7801422733236,    5247950921612,    1948095929808,     308129028480;
  9: 0, 65548320768000, 602727313158144, 2397210664294592, 5397390585784992, 7535880792010329, 6689080734015412, 3689623938848652, 1157135274573264, 158070191610240;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (n^3-3*n^2+3*n-1+(2*n^3-6*n^2+6*n-1)*x+(n^3-3*n^2+3*n)*x^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^3 - 3*n^2 + 3*n - 1 + (2*n^3 - 6*n^2 + 6*n - 1)*x + (n^3 - 3*n^2 + 3*n)*x^2)*P[n - 1, x])/(1 + x)]];
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^3 - 3*n^2 + 3*n - 1 + (2*n^3 - 6*n^2 + 6*n - 1)*x + (n^3 - 3*n^2 + 3*n)*x^2)*v[i - 1])/(1 + x)); 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, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Parent-row provers of polytool 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.