A161198 → basis A013611 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(-1 + 6\,x + 7\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	11														14
3	15	94	143													252
4	105	883	2411	2145												5544
5	945	9732	36710	60292	36465											144144
6	10395	125007	588718	1360702	1546663	692835										4324320
7	135135	1843386	10278481	30052204	48681361	41486778	14549535									147026880
8	2027025	30758895	196575093	687188123	1421421107	1741972973	1172438919	334639305								5587021440
9	34459425	573576840	4110748956	16596575416	41343861894	65149068216	63480785948	34999841160	8365982625							234654900480
10	654729075	11828364435	93590804844	426325154716	1233640912218	2354116567242	2965159774844	2378978202636	1103949381195	225881530875						10794125422080
11	13749310575	267382796310	2308429470339	11666961589512	38269888643342	85212034366404	130537735721742	135948175725832	92173304881539	36758044203030	6550564395375					539706271104000
12	316234143225	6576032942955	61382744503407	339901430139285	1241883248071738	3146243338370894	5643940986958590	7173477849068138	6334379459776189	3702807467997399	1290162351387555	203067496256625				29144138639616000
13	7905853580625	174836550300300	1751577699702690	10523166322094556	42263828396389855	119638230645639704	244924554840204252	365586998796336920	395084255513653279	301604708872549212	154446705228603042	47652045002204940	6701227376468625			1690360041097728000
14	213458046676875	4997291733429975	53411877152483130	345430710186147162	1509434187975835545	4709466221305916933	10800301053282904444	18443208386908245660	23462819856740430733	21971276082536693489	14726225851711504554	6692239898060639850	1848754714241825775	234542958176401875		104802322548059136000
15	6190283353629375	152819407996513650	1733844231558919845	11993730050040143508	56554527728186675799	192423585372977506222	487458980733523155397	934464182191806588568	1362820486161077580677	1505291341092960048302	1239997764753491291159	738945313557084224148	301226762941256621925	75205670214063208050	8678089452526869375	6916953288171902976000

Technical details

Node id: n-8f4e719f262503e9 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-25be57ce791b0e74

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         1;
   2:         3,          11;
   3:        15,          94,         143;
   4:       105,         883,        2411,         2145;
   5:       945,        9732,       36710,        60292,         36465;
   6:     10395,      125007,      588718,      1360702,       1546663,        692835;
   7:    135135,     1843386,    10278481,     30052204,      48681361,      41486778,      14549535;
   8:   2027025,    30758895,   196575093,    687188123,    1421421107,    1741972973,    1172438919,     334639305;
   9:  34459425,   573576840,  4110748956,  16596575416,   41343861894,   65149068216,   63480785948,   34999841160,    8365982625;
  10: 654729075, 11828364435, 93590804844, 426325154716, 1233640912218, 2354116567242, 2965159774844, 2378978202636, 1103949381195, 225881530875;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(4*n+6)*x+(2*n+7)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (4*n + 6)*x + (2*n + 7)*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 1 + (4*n + 6)*x + (2*n + 7)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 basis_map_A013611 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.