A008955 → 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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	7	10													18
3	1	17	80	100												198
4	1	34	369	1460	1700											3564
5	1	60	1253	11054	39660	44200										96228
6	1	97	3473	57415	448658	1511620	1635400									3656664
7	1	147	8323	231065	3319408	23944520	77216400	81770000								186489864
8	1	212	17878	772060	18338633	239706040	1633610200	5100836000	5315050000							12308331024
9	1	294	35262	2238056	81647553	1743473946	21289505480	139056872400	423583602000	435834100000						1021591474992
10	1	395	64956	5799518	307691209	9989876799	197380374026	2289296925880	14468327714400	43217777902000	44019244100000					104202330449184
11	1	517	113146	13724150	1015232405	47528204297	1416145343504	26369702557052	293762552671760	1808353759058800	5316588148144000	5370347780200000				12816886645249632
12	1	662	188111	30130320	3005234155	194736903022	8307734966569	231710777365132	4117369423444300	44403923896464000	267527883211670000	776275629261080000	778700428129000000			1871265450206446272
13	1	832	300651	62109190	8127388555	705626709372	41413008480309	1644025721681862	43508201575516740	744356725881995000	7816194945610550000	46256015775244980000	132745557402512600000	132379072781930000000		319986391985302312512
14	1	1029	464555	121337437	20362898985	2306722254707	180421470226593	9802388392302735	367381268746843554	9315472436258792780	154454469944363565000	1586046420060523330000	9245180665125773660000	26283253881076912200000	26078677338040210000000	63357305613089857877376

Technical details

Node id: n-7b2cbe30a868f205 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-ef50924260c73c5c

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: 1,   2;
  2: 1,   7,    10;
  3: 1,  17,    80,     100;
  4: 1,  34,   369,    1460,     1700;
  5: 1,  60,  1253,   11054,    39660,      44200;
  6: 1,  97,  3473,   57415,   448658,    1511620,     1635400;
  7: 1, 147,  8323,  231065,  3319408,   23944520,    77216400,     81770000;
  8: 1, 212, 17878,  772060, 18338633,  239706040,  1633610200,   5100836000,   5315050000;
  9: 1, 294, 35262, 2238056, 81647553, 1743473946, 21289505480, 139056872400, 423583602000, 435834100000;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (1+(n^2+2)*x+(n^2+1)*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[((1 + (n^2 + 2)*x + (n^2 + 1)*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] = ((1 + (n^2 + 2)*x + (n^2 + 1)*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 of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - 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

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.